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QB Brock Osweiler - Free Agent
6-7, 242 Born: 11-22-1990 College: Arizona State Drafted: Round 2, pick 57 (2012)
2018 Week 4 vs NE (4 / 5 / 35 / 1 / 0 pass, 1 / 7 / 0 rush)
This was probably not the opportunity that Osweiler wanted but he got his shot in relief of Tannehill in the 4th quarter. Osweiler looked good but it all came in garbage time against a soft New England defense that was playing with a big lead. He found Frank Gore on a 3rd and 2 play in the red zone. Osweiler rolled right as Gore leaked out into the flat. Gore broke a tackle and then walked into the end zone. His 1 rush attempt for 7 yards earned him second on the team in rushing this week and that is not something Kenyan Drake owners want to see. Osweiler is not taking over for Tannehill and will be relegated to backup duties again next week against the Bengals.
2018 Week 6 vs CHI (28 / 44 / 380 / 3 / 2 pass, 2 / 8 / 0 rush)
It was a surprise start for Osweiler and an even more surprising victory for the backup quarterback. With 3 touchdowns and nearly 400 yards passing, Osweiler looked like the solid quarterback that had landed him his big money deal previously. He got a lot of help from Albert Wilson and a few really big plays, but he had an all-around solid day behind center. His first touchdown was a 5-yard toss to Nick O'Leary. The big tight end leaked out into the flat after the play action and Osweiler hit him in stride, so he could outrun his man to the pylon for the touchdown. The second touchdown did not come until the 4th quarter and was a product of Albert Wilson more than anything else. It was a quick hitting play to the sideline that Wilson was able to turn up the field and follow his blockers for 43 yards and the touchdown. Osweiler got the ball out quickly and it gave Wilson the chance to get to the second level in a hurry. His third touchdown was of the 75-yard variety and again came to Wilson before a 2-pt conversion to Kenny Stills tied the game late. Osweiler did have back to back drives (albeit sandwiched by half time and a Bears touchdown) where he threw 2 interceptions. The first came after the Dolphins had recovered a Bears fumble and it was a chuck and pray play right before half. There Bears did not have enough time to do anything with the ball and it was just Osweiler trying to make something happen before heading to the locker room. After half, the Bears opened with a touchdown and Osweiler tried to answer. Unfortunately, he stared down his receiver and had the route jumped by the corner. If it weren't for Mike Gesicki it would have been returned for a touchdown. Osweiler got a lot of help from the ground game this week and Albert Wilson, it will be interesting to see if he gets a shot next week or if Tannehill will regain his starting spot.
2018 Week 7 vs DET (22 / 31 / 239 / 2 / 0 pass, 1 / 2 / 0 rush)
After setting a career high in yards last week, Osweiler followed it up with another solid performance. He did not turn the ball over and helped to move the chains with short passes to Amendola and Drake. He combined to throw 15 of his 31 passes to these two and included one of his touchdowns. The first touchdown pass came on a 5-yard fade route to Kenny Stills. The speedy receiver made his way to the corner of the end zone and Osweiler led him perfectly where only he could make a play on the ball. The pass led Stills so far that he ended up running into a security guard and knocking her over. The second pass was another perfectly thrown ball to Amendola for 34 yards. Osweiler rolled right and threw a ball just over a diving defender and before the crashing safety could get over to make the play. The biggest issue this week was the pass protection as Osweiler was sacked 4 times and lost 24 yards. There has been no timetable set for when Tannehill will return, but ownership has been quoted saying that it will be his team again when healthy. With the short week, do not be surprised to see Osweiler starting again against the Texans.
2018 Week 8 vs HOU (21 / 37 / 241 / 0 / 1 pass, 0 / 0 / 0 rush)
The Dolphins came out swinging this week, but Osweiler could not deliver. They turned the ball over on downs the first drive after he failed to connect with Frank Gore on a 4th and 1 play. They then got the ball rolling with a touchdown after a penalty nullified a field goal and gave the Dolphins a fresh set of downs. It was a rushing touchdown and Osweiler played little role on that drive. The third drive resulted in an interception and I think you can see a trend forming here. Osweiler was then buried in a hole behind a career game from Deshaun Watson and he did not even get a chance to put up garbage time stats. The lone passing touchdown came from Danny Amendola rather than Osweiler. The longest connection of the game was a 46-yard pass to DaVante Parker and even after that the Dolphins settled for a field goal. Osweiler was sacked twice and just did not seem to have his best stuff this week. There is talk that Tannehill will be back next week and that should end the short tenure as a starter for Osweiler. Barring another injury, he will remain on the bench the rest of the season.
2018 Week 9 vs NYJ (15 / 24 / 139 / 0 / 0 pass, 3 / 4 / 0 rush)
It was not the best showing for Osweiler, but he did just enough to get his team the win this week. He finished without any touchdown passes and only completed 62% of his passes. He targeted Amendola and Drake for just over half of his passes and failed to get the tight ends involved this week. His longest play of the game was a 26-yard catch and run play to Amendola and another 19-yard pass to Stills. He was sacked 4 times and failed to move the chains when it mattered most. He led the offense to 2 field goals and was very thankful for an interception returned for a touchdown. This was the difference in the game and it had nothing to do with Osweiler. There is a chance that Osweiler only gets one more start for the Dolphins before Tannehill returns after the Bye. He has sprinkled in some very good games and some duds in his 4 starts so far this year. Look for another tough matchup in Green Bay next week.
2018 Week 10 vs GB (23 / 37 / 213 / 0 / 1 pass, 1 / 0 / 0 rush)
Any sense of magic or talk that Osweiler may have found the ability that caused Houston to give him a monster deal a few years ago should be dead after this week. The Packers shipped off their top safety and have struggled on defense all season long, yet the Miami offense only managed 4 field goals. Osweiler continues to not be able to finish drives and is forcing his throws. He was fixated on Amendola and Parker this week and targeted the pair on almost 2/3 of his attempts. He seems unwilling to target his tight ends with both Gesicki and O'Leary seeing 1 target for 5 yards each, and not a single target to Grant in the entire game. Osweiler may not have the best weapons in the NFL, but he certainly has more options than he is using. His lone big play of the game was a 39-yard catch and run by Amendola. All of the credit goes to Amendola who cut up field and through the void left by the zone coverage. The interception was a poor read by Osweiler. He stared down Amendola on a short route and did not see Breeland break on the ball. He effectively threw a quick pass into double coverage and paid for it. Initial reports said Tannehill would be back after the Bye week, but the most recent said that it may be Osweiler for the rest of the season. This may be the last real audition that he gets, and he will have to play much better if he even wants to continue to be a backup in the NFL.
2017 Week 9 vs PHI (19 / 38 / 208 / 1 / 2 pass, 0 / 0 / 0 rush)
Denver, seeking direction after a rough few weeks, turned to Osweiler at quarterback. The results were not spectacular, as the QB was unable to establish a rhythm with any receiver other than Thomas. Osweiler's longest connections of the day were a 30 yard corner route to Sanders on play action in the 2nd, and a 32 yard bomb down the left sideline to Cody Latimer at the end of the 3rd. Osweiler force-fed Demaryius Thomas slants, curls, and bubble screens throughout the contest. Thomas and Osweiler finally scored in the 4th quarter on a WR bubble screen from the Philadelphia 2 yard line. Osweiler nearly threw an early 1st quarter interception while looking for Sanders on a curl route, but the ball bounced off the defenders' chest. Osweiler made an awful decision later in the 1st, throwing the ball right to another defender who was covering Thomas step-for-step on a slant route. This time the pass stuck, and the Eagles returned the ball to the Denver red zone. Osweiler came right back out on the next drive and slung a 3rd-and-long pass into triple coverage, nearly resulting in another pick. Osweiler threw a high pass to Booker on a crossing route in the 3rd, nearly resulting in another pick as Booker couldn't make a leaping grab and deflected it to an Eagles defender. Osweiler finally connected with another Eagle in the 3rd quarter. Staring down a free-blitzed, Oz chucked the ball straight up in the air, in the general direction of a streaking Demaryius Thomas. The ball landed about 20 feet away from Thomas, right in the hands of the free safety. Despite the slow performance, the Broncos claim they will stick with Osweiler in week 10.
2017 Week 10 vs NE (18 / 33 / 221 / 1 / 1 pass, 1 / -1 / 0 rush)
Poor special teams play early in the contest dug the Broncos into a hole that Osweiler just wasn't able to pull them out of. Osweiler led a couple of scoring drives early, leaning on a solid connection with Emmanuel Sanders. Unfortunately, Denver couldn't find the end zone and settled for field goals while the Patriots racked up touchdowns. Osweiler finally found Demaryius Thomas for a touchdown late in the 3rd quarter, dropping back into the pocket while Thomas boxed his defender out on a crossing route for the score from the New England 7. Osweiler was intercepted by safety Patrick Chung late in the 4th after making a questionable throw to TE Heuerman, who was double covered on an out route.
2017 Week 11 vs CIN (23 / 42 / 254 / 1 / 1 pass, 4 / 24 / 0 rush)
Osweiler was terrible in the first half, throwing one interception and racking up less than a hundred yards with no touchdowns. Oz had a nifty 12 yard scramble to convert 3rd and short in the 1st quarter, setting the Broncos up inside the Cincinnati red zone. Unfortunately, he hit CB Dre Kirkpatrick in the end zone two plays later, leading to a 99 yard INT return and an ensuing Cincinnati touchdown. CJ Anderson tossed the ball back to Osweiler after a handoff for a 3rd quarter flea-flicker, but there was apparently a miscommunication on the play as Osweiler launched the ball to a very empty spot downfield. Osweiler found a rhythm with Demaryius Thomas in the 4th quarter, hitting him for several passes before capping the drive with a 17 yard touchdown on a beautiful over-the-shoulder fade. Osweiler and the offense got the ball back with just under two minute left in the game, trailing Cincinnati by three points. Osweiler got blown up on 2nd down, leading to 3rd and 17 with less than a minute left, then missed Sanders on a slant route on 4th and short. Osweiler finished the day with three official sacks for 25 yards.
2017 Week 15 vs IND (12 / 17 / 194 / 2 / 0 pass, 3 / 17 / 1 rush)
Osweiler was pressed into service after Siemian injured his shoulder at the end of the 1st quarter. Osweiler ran the play-action passing game to perfection behind Anderson's strong performance on the ground. The offense went three-and-out for his first series, but Osweiler led a scoring drive at the end of the 2nd quarter, capping it with an 18 yard touchdown run. The Colts ran man coverage with no one spying on Osweiler, and he barreled up the middle of the field to the end zone. Osweiler's first touchdown pass was a beautiful corner route to Cody Latimer, who beat his man from the left slot in a spread formation. Osweiler took the shotgun snap and lofted the the ball over the defender. Osweiler found Jeff Heuerman later in the 3rd quarter on a play-action post route that the tight end took to the house for a 54 yard score.
2017 Week 16 vs WAS (22 / 38 / 193 / 0 / 1 pass, 4 / 27 / 0 rush)
Osweiler had a nice rhythm going with Demaryius Thomas on the opening drive, hitting him down the seam for a big gain on the first offensive snap, then an out route for a nice catch-and-run. The Denver passing game quickly devolved into a complete mess as Osweiler was constantly harried by pressure from the Washington front 7. Sanders and Latimer were out, and Osweiler couldn't get anything going downfield. Osweiler targeted his backs 20 times (Anderson-9, Booker-8, Janovich-2, Henderson-1) on passes that were mostly checkdowns or screens. Osweiler fumbled and lost the ball while attempting to escape a Ryan Kerrigan sack late in the 2nd quarter. The Broncos got the ball back before the half, but Osweiler unfortunately tossed an interception after the ball was tipped at the line of scrimmage. Osweiler was sacked four times and was mostly ineffective with his scrambling, typically sliding down a yard or two past the line of scrimmage. He did break one long run of 18 late in the 4th quarter after Washington had moved to a soft prevent defense with a 24 point lead. Osweiler and the offense led a scoring drive late in the 4th, with CJ Anderson punching in a five yard touchdown and catching a short two-point conversion. Denver has announced that Paxton Lynch will start in week 17.
2016 Week 1 vs CHI (22 / 35 / 231 / 2 / 1 pass, 4 / 14 / 0 rush)
It was a productive day for Osweiler on the day and it started with some big throws down field to DeAndre Hopkins. He was focused on Hopkins early in the game and threw a good ball in traffic that Hopkins got his hands on but was ripped away by a Bears cornerback. It is a ball that Hopkins usually comes up with but this one goes against Osweiler but it was not entirely his fault. Looking to Hopkins, he ended up hitting him for a 23 yard touchdown strike in the end zone on a 9 route. Hopkins ran a nice route and Osweiler put some air under it to let him go to work. Osweiler would have had an 80+ yard touchdown pass if Will Fuller was able to haul it in when he split two defenders getting verbal. Oswieler showed some faith in the youngster and went to him in the second half hitting him from times for 100 yards. He checked into a player to hit Fuller on a smoke screen for 18 yards and touchdown where he went untouched to the end zone with some good blocking. There was a crucial drop on a well place ball in the end zone to Hopkins from Osweiler that should have been a touchdown. He worked underneath to Braxton Miller, Lamar Miller and the tight end group but he did most of his work with Hopkins and Fuller. He tried to push the ball down field late to Fuller to deliver the knock out blow but could complete three attempts of 20 yards or more. Osweiler showed he is not afraid to push the ball down field and let his receivers work.
2016 Week 2 vs KC (19 / 33 / 268 / 1 / 2 pass, 2 / -3 / 0 rush)
It was a mixed bag for Osweiler and it came due to inaccurate throws with no real pressure. His two interceptions were poorly thrown footballs, both were behind their intended targets. One was behind Braxton Miller in the end zone and the other was behind DeAndre Hopkins on a stop route thrown behind him that Hopkins got one of his hands up but popped right into the hands of Marcus Peters. He is a true gunslinger and has a short term memory and continued to throw the football. After the interception in the end zone, he came back the next offensive series and hit Hopkins for a 27 yard touchdown pass in-between two Chiefs defenders. He was more effective in the middle of the field hitting Fuller for gains of 14 and 19 yard gains and Hopkins for 25, 16 and 14 yard gains. Last week Osweiler did not attack the intermediate portions of the field but he did this time and it paid off. Osweiler is a heady player and will take advantage of mishaps from the defense and he hit Jonathan Grimes on a key 13 yard gain late in the game with a kick swing pass outside the numbers when the Chiefs defense could not get lined up correctly. Osweiler's completion percentage will remain low due to the fact he just can't say no to throwing the ball deep. He trusts Will Fuller completely to change the dynamic of the game and he is going try for knockout punches with deep balls. The big play ability is there for Osweiler but he is still learning the offense and until the running game picks up, it will all be on Osweiler to provide most of the offense with his arm.
2016 Week 3 vs NE (24 / 41 / 196 / 0 / 1 pass, 1 / 13 / 0 rush)
By the time Brock Osweiler and the Texans figured out the Patriots defense, it was all but too late. Osweiler received his first real taste of being forced to being patient. His big plays were non-existant that he showed the first two weeks. It fell on Osweiler to work short and intermediate routes and he had some success to DeAndre Hopkins hitting him five times for 56 yards all on routes to the right side of the field. The one interception Osweiler did throw was in a tight window intended for Hopkins in the middle of the field, that Jaime Collins stepped in front of. He also had a 20 plus yard gain wiped out due to another Will Fuller drop which would have helped the cause. The Texans offense was shrunk down to check downs and crossers less than 10 yards and tight end Ryan Griffin ended up being his favorite target late in the game catching 8 passes late in the game. Osweiler was impatient all night and throwing the ball 41 times, there were plenty of yards left on the field for the offense and Osweiler.
2016 Week 4 vs TEN (25 / 37 / 254 / 2 / 2 pass, 1 / 14 / 0 rush)
His day will clouded with his two poor interceptions both intended for DeAndre Hopkins. One was forced into coverage when he was forced up in the pocket and he panicked and threw it blind to Hopkins. His other interception was the right read but he threw it off his back foot with pressure coming and left it short with Hopkins clearly past the the defender. Osweiler can not stay away from the interceptions and it ruined what was his best day passing of the season. Osweiler was efficient and did majority of his work underneath and let his receivers work. He threw good passes that allowed his receivers to catch and eat up yards. He hit Ryan Griffin on a crossing route for 45 yards to set up a score early in the first quarter. He had two touchdown throws one to C.J. Fiedorowicz where he looked off the safety to hit him up the seam for a 14 yard touchdown for a leaping reception. Osweiler continues to look Will Fuller's way and too advantage of his speed on a smoke screen that went for 28 yards and he hit him for a 5 yard touchdown play after a good play fake that Fuller sold as a run. Osweiler spread the ball around and hit 8 different receivers and finally started looking to the middle of the field where the tight ends and Jaelen Strong caught five passes to keep the chains moving. It was a solid production day for Osweiler but he has to find how to cut down on his interception problem.
2016 Week 5 vs MIN (19 / 42 / 184 / 1 / 1 pass, 0 / 0 / 0 rush)
He continues to be a mystery as a player and his play against the Vikings was a mess. Once again Osweiler had an interception where he threw it intended for Jaelen Strong where he was not even looking for the football. Osweiler was under pressure more than normal and it comes from a running game that was nonexistent against the Vikings. Osweiler threw the ball 42 times and only completed 19 of his passes with his longest throws going to DeAndre Hopkins on an out route that Hopkins broke a few tackles for a 25 yard gain. He hit C.J. Fiedorowicz for a 22 yard gain and it could have easily been a touchdown if Osweiler would have led him. Fiedorowicz had the linebacker beat and had to make a leaping catch to haul in the high pass. Osweiler had his only touchdown of the game on the Texans final drive of the game hitting Hopkins for a 1-yard touchdown on a quick slant. The game was well out of hand when Osweiler threw his touchdown but Osweiler's inaccurate passing is a growing concern.
2016 Week 6 vs IND (25 / 39 / 269 / 2 / 1 pass, 2 / -4 / 0 rush)
This was a tale of 57 minutes of Brock Osweiler and the final three minutes and overtime version of the Texans quarterback. Osweiler continued to look uneasy in the pocket to start the game and he could not get comfortable in the pocket and sailed passes early in the game intended for DeAndre Hopkins. Later in the game he left the ball behind Hopkins on crossing route and threw it right into Vontae Davis hands for another interception. In the first three quarters Osweiler went 11 of 22 for 89 yards with the interception. Then when it became time for the Texans to make plays, Osweiler's light switched on. Osweiler starting getting into ta groove by working underneath to Ryan Griffin for 12 yard gains. His first touchdown came on a check down to Lamar Miller for a 16 yard touchdown that saw him beat seven defenders with some help for the touchdown. Then went to his favorite target over the past few weeks in C.J. Fiedorowicz and capped off their comeback with a 26 yard strike up the seam for a touchdown on a perfectly thrown ball. Osweiler also finally started picking away at the Colts defense by working to DeAndre Hopkins with 8-10 yard chunks. His best ball came to Jaelen Strong on a 9 route for 36 yards in overtime to set up the game winning field goal in overtime.
2016 Week 7 vs DEN (22 / 41 / 131 / 0 / 0 pass, 3 / 16 / 0 rush)
It was an ugly day for Brock Osweiler and the worst part of it is that the offensive line have him time to work throwing the football. His longest completions of the night went for 13 to Will Fuller who helped him out on a catch and run. He was inaccurate all evening and could not get into a rhythm throwing the football. Osweiler lost a fumble that was a questionable call that could have been ruled a pass, that alone helped sink the Texans late in the game. There was not much to feel good about Osweiler's performance that saw him work underneath majority of the night and very little down field. Osweiler have 5 of his 41 passes go for more than 10 yards all game. He tried 4 passes of 20 yards or more which all came up empty. Osweiler did have three key scrambles for 16 yards to extend drives and his longest run was 10 yards on a key third down. Until Osweiler figures out to sit solid in the pocket after his first read is taken away or his receivers get off of press coverage, it will continue to be tough for Osweiler to work.
2016 Week 8 vs DET (20 / 29 / 186 / 1 / 1 pass, 3 / -3 / 0 rush)
There are positives about Brock Osweiler's day and it starts with it was easily his most accurate day as a passer in a Texans uniform. Osweiler attempted two deep passes which was his lowest of the season and one was just out of bounds in the end zone to DeAndre Hopkins. Osweiler had five drops by his receivers, which dropped his numbers and only really had three pass they were really poor throws. He conintues to work well with his tight ends and he hit Ryan Griffin for a 23 yard strike on a 3rd down up the seam and he continues to work well with C.J. Fieodorwicz who he hit for another touchdown. On the 6 yard touchdown throw, Fiedorowicz ran and out route underneath the void Hopkins created which was an easy pitch and catch. Osweiler was much better on the intermediate part of the field and his only interception came on a 3rd down throw to Hopkins where he expected him to cut off his route for a back shoudler throw. His stats will not say it but Osweiler's most efficent day came from a controlled passing game helping him out get some good looks in the middle of the field.
2016 Week 10 vs JAX (14 / 27 / 99 / 2 / 0 pass, 3 / 23 / 0 rush)
The game dictated on what the Texans were doing with the football and Brock Osweiler did the most important thing, no turnovers. Osweiler protected the football and worked the middle of the field to the tight ends where he had his most success. He only had two passes targets over 20+ yards the entire game both for Hopkins but he were well covered. Osweiler continued to find success to the tight end group. Both of his touchdowns came on plays from the one yard line, one on boot leg hitting Ryan Griffin for an easy pitch and catch. Also, hitting Stephen Anderson on another play action pass for a simple one yard touchdown. Osweiler is still struggling to find DeAndre Hopkins consistently, some can be attribute to poor passes but some can be pointed to Hopkins not getting into routes. Osweiler's biggest pass came to ice the game hitting Hopkins on a 10 yard slant to win the game. There was not much happening in the passing game with the Texans who played with a lead majority of the game and did not trail at any point. It was Osweiler's lowest attempts of the season and he made sure he did not force the ball during the game.
2016 Week 11 vs OAK (26 / 39 / 243 / 1 / 1 pass, 2 / 19 / 0 rush)
Most will not discuss it but it has been a solid three game stretch for Brock Osweiler. Osweiler had a 70+ yard touchdown reception called back when DeAndre Hopkins had a 24 yard reception on a crossing route that he clearly did not step out of bounds and the officials blew the play did thinking he did. Osweiler completed 26 passes and it was his most efficient game of the season. He hit 8 different receivers and C.J. Fiedorowicz remained his favorite target on the field hitting him 6 times and mainly in the middle of the field in catch and runs. Osweiler did have two near interceptions that hit Raider defenders in the hands but they couldn't finish the play. Osweiler also started getting rolling with Hopkins out of the slot, and hit him for gains of 13, 9, 8 and 4 yards. Osweiler's only touchdown of the day came on a pass to Braxton Miller for a 12 yard touchdown reception hitting him with an easy throw from the slot. Osweiler was under control the entire night and it is clear he is settling into his role with the offense.
2016 Week 12 vs SD (22 / 37 / 246 / 0 / 3 pass, 3 / 23 / 1 rush)
Maybe one game the interceptions will go away but Brock Osweiler continued to struggle to protect the football. He did not get much help from his offensive line who turned in one of their worst performances of the season in pass protection and run blocking. Osweiler worked intermediate all game and had success throwing to ball to C.J. Fiedorowicz in the middle of the field. He finally looked comfortable with DeAndre Hopkins and Osweiler made sure he did what he could to get him the football. Osweiler his Hopkins five time for 70 yards and did the two did most of their work in man coverage situations. Osweiler did have a rushing touchdown on a sneak for one yard but that was the highlight of his game. He had two interceptions that were on him, one ball that he should have thrown away but forced it to Hopkins. The other came on an overthrow to Fiedorowicz who was wide open down the seam that ended up in the safeties hands. His final interception came on a hail mary to end the game but his accuracy issues for the game are still coming back to bite him. There are moment of good for Osweiler but the bad decisions are clouding and stunting his growth as a quarterback.
2016 Week 13 vs GB (22 / 35 / 202 / 2 / 0 pass, 3 / 20 / 0 rush)
Patience will continue to be the key for Brock Osweiler and he continues to have solid games mixed in. This was a bounce back performance after a horrid game last week but this time in poor conditions at Lambeau field. Osweiler threw for 202 yards with no turnovers and two touchdowns. He was also not helped about by four drops from receivers and two coming from his most dependable target in DeAndre Hopkins. The most crucial coming on 3rd down play late in the game. Osweiler continued to spread the ball around and hit seven different receivers with Will Fuller and Hopkins being the bigger beneficiaries. Osweiler his Hopkins late in the game against cover two with a solid 44 yard touchdown pass that saw Hopkins with the long catch and run. Osweiler worked underneath majority of game and hit Fiedorowicz 6 times and Griffin 4 times. His first touchdown pass went to Griffin on a key 4th down play for 6 yards in the corner of the end zone on a play action pass. Osweiler also rushed for 20 yards in key spots to convert two 3rd downs. It was an overall solid performance for Osweiler, especailly after they opened up the offense in the second half and starting pushing the ball downfield.
2016 Week 14 vs IND (14 / 24 / 147 / 0 / 1 pass, 2 / -2 / 0 rush)
It continues to be game managing 101 for Brock Osweiler but some of this is put on the coaches and their game plan. Osweiler only had two downfield throws and once again worked intermediate once again to C.J. Fiedorowicz. Fiedorowicz also dropped two passes and Will Fuller dropped on a tough deep ball down the sideline that was in his hands. Osweiler also missed Hopkins on a deep ball to open the game that was just out of his reach down the sideline. He spread the ball out to 8 different receivers and his longest pass went for 22 yards to Hopkins. His one interception came on a play call where he did not see the linebacker drop under Fiedorowicz and threw it right to him. He continues to take the easy throws and leaves it up to his receivers to make plays.
2016 Week 15 vs JAX (6 / 11 / 48 / 0 / 2 pass, 0 / 0 / 0 rush)
Osweiler only attempted 11 passes and two interceptions later, Bill O'Brien saw enough. He did hit Ryan Griffin twice for gains of 9 and 7 on play action passes but his interceptions did him in. He had a pass thrown too high for Will Fuller that probably could have been caught but went through his hands into a defenders hands. Then Osweiler scrambled out of the pocket targeting DeAndre Hopkins but threw it right to an inside linebacker sitting right in front of him. That was the end of the game for Osweiler after the interception which led to Tom Savage.
2016 Week 17 vs TEN (21 / 40 / 253 / 1 / 0 pass, 1 / 1 / 1 rush)
Osweiler came in relief of Savage off the bench side second quarter and put together a respectable game. Some issue up front in protection led to some hard hits on Osweiler but he stuck in there and made some solid passes. He finally got DeAndre Hopkins going and hit him for over 100 yards in the 3rd quarter alone on 5 passes. He hit Wendall Williams for a 26 yard strike on the sideline to open up the offense. Osweiler hit C.J. Fiedorowicz for a short score to end a nice drive in the middle of zone coverage. He also ran one in on 4th down late in the game that saw him use a spin move to get into the end zone. Osweiler did miss some open throws in the red zone to receiver that were wide open one to Stephen Anderson and another to Keith Mumphery.
It was clearly Osweiler's best half of football for the Texans this season. Only asked to throw it 7 times in the second half, Osweiler was busy early in the game hitting Hopkins on a deep ball for 38 and Fuller on a solid post route for 19. Osweiler had plenty of time to operate and make solid passes and most importantly his accuracy looks much improved than games before. His best play came on the goal line on a bootleg where he had a 1 yard touch down run shaking a Raiders defender to the ground going into the end zone untouched. Completing over 67% of his passes in the first half Osweiler did not turn the football over and it is clear that he was a different quarterback this game.
2016 Week 19 vs NE (23 / 40 / 198 / 1 / 3 pass, 1 / 18 / 0 rush)
The stat sheet shows a complete meltdown but the three interceptions came late in the game when the Texans needed to start airing out the football. What is lost in that in the first half, Osweiler had three well placed footballs in the end zone not hauled in. C.J. Fiedorowicz stumbled and could not haul in a pass that hit his finger tips. Will Fuller dropped a post route dropped perfectly over his shoulder and the ball slipped right through his hands to the ground right before the half. Then add another ball thrown down the right sideline that was a 50/50 ball that Fuller only got his right hand on but was also a well placed ball for only Fuller to make a play on. That lack of help slowed Osweiler's day but he did have a touchdown pass to C.J. Fiedorowicz on a play action pass that he hit wide open in the end zone. Osweiler played under control but if he would have gotten some help from his receivers, the game might have taken a different turn.
2015 Week 10 vs KC (14 / 24 / 146 / 1 / 1 pass, 3 / 18 / 0 rush)
While starter Peyton Manning's early-season struggles had led to calls from fans and analysts for Denver to see what Osweiler could do, few expected the unproven backup to receive his chance so soon. But with Manning unravelling on the field in front of everyone, Denver had no choice but to put Osweiler in against the Kansas City Chiefs. Osweiler looked like every bit of the inexperienced player that he is, throwing an interception shortly after coming in and heaving a few other errant and dangerous throws. He also showed off his surprising athleticism for a man of his height, rushing for 18 yards on three scrambles and providing enough of a spark to drive a stagnant offense down for two face-saving touchdowns against a long-since-disinterested defense.
2015 Week 11 vs CHI (20 / 27 / 250 / 2 / 0 pass, 2 / 4 / 0 rush)
For the first time in his four-year career, Brock Osweiler recorded a first-quarter snap for the Denver Broncos. As was to be expected given the long-delayed nature of his debut, Osweiler's offense produced in fits and starts. Head coach Gary Kubiak simplified the passing game, not asking him to throw a single pass more than 15 yards downfield and focusing heavily on slants, screens, and rollouts. 14 of Osweiler's 20 completions went to running backs or tight ends. Of Osweiler's 89 yards to wide receivers, 48 game on a short pass over the middle to a wide-open Demaryius Thomas, who did all of the hard work after the catch. The offense's inability to push the ball downfield meant drives almost always stalled out after a single negative play. At the same time, Osweiler showed positive signs, too. Denver's offense moved the ball better than its 17 points would suggest; an 82 yard drive ended in a short field goal on 2nd-and-goal as the first half expired, and Denver failed to convert on 4th-and-1 from Chicago's 2 yard line. Osweiler's ability to line up under center helped the running game get going, (against an admittedly soft defensive front). For the first time all season, Denver's quarterback did not throw an interception, (and its offense as a whole did not commit a turnover). While an up-and-down first start, there were positive aspects to build upon.
2015 Week 12 vs NE (23 / 42 / 270 / 1 / 1 pass, 2 / 4 / 0 rush)
Brock Osweiler's first career start was a win on his birthday. In his second, he knocked off the undefeated Patriots in front of a friendly home crowd. While Osweiler's inexperience shone through on multiple occasions, the lasting memory most will take was him showing poise and precision in a do-or-die situation against a punishing defense. Getting the ball down by 4 with 2:31 to go, Osweiler engineered a 6-play, 83-yard drive for the go-ahead touchdown. 79 of those yards came through the air, via 36- and 39-yard completions and a 4-yard touchdown toss. While Osweiler came up big with the game on the line, his performance prior had plenty of highs and lows. While Osweiler's lone interception was in many ways not his fault, (his hand was hit as he began to throw and the ball popped straight up into the air), it was emblematic of his poor feel for the pocket. Osweiler took a pair of completely inexcusable sacks; in one he ran backwards from the pressure on 3rd-and-2, losing 13 yards and knocking Denver out of easy field goal range. On his phenomenal 4th-quarter drive, too, Osweiler ran backwards from pressure and took an 8-yard sack on 2nd-and-goal; fortunately, the latter was negated by defensive holding. Overall, Osweiler showed plenty of reason for optimism that Denver's offense will be able to hold its own going forward, while also demonstrating areas for growth in the future.
2015 Week 13 vs SD (16 / 26 / 166 / 1 / 1 pass, 3 / 1 / 0 rush)
When Brock Osweiler led the Denver Broncos to a touchdown on their opening drive against the San Diego Chargers, it seemed as if he was poised to build upon his success against New England. Instead, Denver's offense became more and more mired the longer it played, especially in the second half as the teams battled to a scoreless draw. Denver once again limited Osweiler's responsibilities, giving him few shots down the field. On one of his rare opportunities, Osweiler lobbed a pass deep down the left sideline to a covered Emmanuel Sanders; the pass, thrown short, was intercepted in the end zone to end a rare promising drive. Osweiler also poorly managed the 2-minute drill to end the first half, almost throwing an easy interception that was dropped by the defense and then dumping the ball over the middle as time expired on the half. On the whole, Osweiler looked much more like the inexperienced quarterback he is than he had in his first two weeks.
On a day where Denver's defense played out of its minds, the Broncos merely needed the offense to play decent football to coast to victory. Unfortunately, (with blame to fall on a terrible running game and a few key late-game drops), Brock Osweiler's inability to convert yards into points in the first half wound up costing the Broncos dearly in the second. He's simply not at a point in his career where he's able to carry an offense on his shoulders. Osweiler had several good throws and positive plays, but by and large had absolutely no feel for the pocket, leading to a career day for the Raiders' Khalil Mack, who had 5 sacks, each one of which ended a Denver drive, and one of which resulted in a safety. Despite being one of the tallest quarterbacks in history, Osweiler also had several batted balls. In the first half, Denver had drives of 10 plays and 57 yards, 9 plays and 61 yards, 15 plays and 63 yards, and 12 plays and 61 yards, all of which ended in red-zone, (or near-red-zone) field goals. The failure to reach the end zone wound up haunting Denver, whose defense allowed one 80-yard touchdown drive and just 49 yards on 12 other drives, (not counting kneel-downs), but lost to an Oakland team that converted its few opportunities into touchdowns.
2015 Week 15 vs PIT (21 / 44 / 296 / 3 / 1 pass, 5 / 19 / 1 rush)
Since Brock Osweiler's ascension to the starting quarterback position, his Denver Broncos have led the league in 3-and-out offensive drives. Against the Raiders last week, the offense's inability to score a touchdown on any of their three red zone trips likely cost them the game. For the first half against the Pittsburgh Steelers, though, the script could not have possibly been any more different; after a fumble by Ronnie Hillman on their second offensive play, Osweiler and the Broncos scored touchdowns on all four remaining drives in the first half. More than just scoring touchdowns, the Broncos were 8-for-8 on 3rd down conversions in the first half, and Osweiler was the primary driver of that. He converted six first downs with his arm, including a 3rd-and-2 throw from an empty backfield that displayed a lot of trust from the coaching staff. And when all his receivers were covered, Osweiler ran for a 7-yard touchdown on 3rd-and-goal. Any hopes that the offense had turned a corner were short-lived; as has been the case in most of his games, the defense seems to have figured out Osweiler at halftime, leading to failed attempts on Denver's first six 3rd-downs in the second half and 8 of 9 total, (plus two more failed 4th-down attempts), as the suddenly listless offense was shut out in the second half. Osweiler threw an interception on a crossing route and had two more easy interceptions dropped by defenders late after Pittsburgh had finally clawed back into the game. And, for the second consecutive week, Osweiler was let down by terrible drops in high-leverage situations. But at the end of the day, despite occasional flashes of talent, Osweiler still looks like he struggles most of the time to get Denver's offense out of neutral without heavy support from the running game.
2015 Week 16 vs CIN (27 / 39 / 299 / 1 / 0 pass, 3 / 5 / 0 rush)
Osweiler continued recent trends against the Cincinnati Bengals; he made safe decisions and took fairly good care of the football, he avoided any killer turnovers, and he continued to show little feel for the pocket, resulting in several drive-killing sacks. With a duo of receivers good enough to make his life easy, Osweiler's play was enough to win against Cincinnati, but still resulted in far too many short drives and quick punts. Even Osweiler's longest play of the game came with an asterisk; Emmanuel Sanders was well behind the Bengals' defense, but had to turn around and wait for a ball that was so high and so underthrown that it looked like he was fielding a punt, turning what could have been an easy touchdown at the two-minute-warning into an eventual field goal as the first half expired.
Brock Osweiler's statistics against the San Diego Chargers tell the story of a player with extreme highs and lows. In reality, his play was pretty middle of the road; he didn't move the ball as well as his 10.5 yards per attempt suggest, nor was he as careless with the football as his 2 interceptions and lost fumble imply. Instead, Osweiler's statistics were padded by a pair of huge catch-and-run plays by his starting wide receivers that accounted for 118 of his 232 yards. At the same time, his turnovers came on a pass tipped by a defender, a pass tipped by the intended receiver, and a lost fumble after a blindside sack. While Osweiler played a decent game against a weak and wounded defense, the quick hook Denver gave him in the second half in favor of Peyton Manning suggests the Broncos believe Manning gives them a better chance to win going forward.
2013 Week 4 vs PHI (2 / 3 / 10 / 0 / 0 pass, 2 / 3 / 0 rush)
Brock Osweiler finally received his first action of the season when the Broncos were up by 36 points against Philadelphia. He had limited opportunities, but he managed to show reasons for guarded optimism while simultaneously reminding that he is not Peyton Manning. He did a very nice job avoiding pressure and making plays with his legs, dancing out of a sack to convert a 3rd-and-3 on the ground. He also managed to guide Denver to a field goal to break the franchise record for points in a single game, and on Denver's final drive Osweiler converted two of Denver's three first downs with his arm as the Broncos ran out what was left of the clock.
2013 Week 17 vs OAK (9 / 13 / 85 / 0 / 0 pass, 1 / -1 / 0 rush)
With records set, a 31-point lead, and the playoffs looming, Denver rested most of its starters in the second half, giving Brock Osweiler a chance to see his first significant game action since week 4 against Philadelphia. Osweiler took good care of the football and managed to move it fairly well, with just one three-and-out (after an amazing reception by Jacob Tamme was overturned on replay), but an offense that scored 31 points on 5 possessions with Manning under center ultimately scored just 3 points on 5 possessions after halftime. Osweiler has shown improvements in his decision making and mobility since his first year, but still tended to hold the ball too long at times and lacked Manning's decisiveness with the football. Ultimately, Osweiler played like a young quarterback with potential and signs of progress, but one not yet ready to keep an offense consistently executing at a high level.
2012 Week 17 vs KC (2 / 4 / 12 / 0 / 0 pass, 3 / -6 / 0 rush)
The rookie from Arizona State took over for Peyton Manning in the fourth quarter, as the Broncos sought to run out the clock as fast as possible. Osweiler completed two of four passes - all attempts were on third down - for twelve yards.

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Home / Embassies / United States to participate in AMAN-13 exercise
United States to participate in AMAN-13 exercise
in Embassies March 6, 2013 254 Views | Short URL
Islamabad, March 06, 2013 (PPI-OT): USS William P. Lawrence (DDG 110), a U.S. Navy Arleigh Burke-class destroyer, has arrived in the waters off the port of Karachi to participate in AMAN-13, a multinational exercise being hosted by the Pakistan Navy March 4-8.
USS William P. Lawrence is a state-of-the-art naval vessel, one of the most advanced in the U.S. fleet, capable of ensuring the free passage of the seas for all nations.
The United States is one of 13 countries with ships actively participating in AMAN-13, whose objective is to promote peace and stability in the region and enhance interoperability among partner naval forces. Twenty other nations are also sending observers to the exercise.
Alberto Rodriguez
Public Affairs Section
U.S. Embassy, Islamabad- Pakistan
Cell: +92300 501 2640
Web: http://islamabad.usembassy.gov
Islamabad- Pakistan U.S. Embassy 2013-03-06
Tagged with: Islamabad- Pakistan U.S. Embassy
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text:
\section{Introduction}

Game theory provides the natural toolkit for reasoning about systems of
multiple self-interested agents.  In some cases, the game is exogenously
determined and all that is left to do is to figure out how to play it.  For
example, if we are trying to solve heads-up limit Texas hold'em poker (as
was effectively done
by~\citet{BBJT15a}), there is no question about
what the game is.  Out in the world, however, the rules of the game are
generally not set in stone.  Often, there is an agent, to whom we will
refer as the {\em designer} or {\em principal}, that has some control over
the game played.  Consider, for example, applications of game theory to
security
domains~\citep{Pita08:Using,Tsai09:IRIS,An12:PROTECT,Yin12:TRUSTSAIMAG}.  In
the long run, the game could be changed, by adding or subtracting security
resources~\citep{Bhattacharya11:Approximation} or reorganizing the targets
being defended (roads, flights, etc.).

{\em Mechanism design} constitutes the extreme case of this, where the
designer typically has complete freedom in choosing the game to be played
by the agents (but still faces a challenging problem due to the agents'
private information).  However, out in the world, we generally also do not
find this other extreme.  Usually, some existing systems are in place and
place constraints on what the designer can do.  This is true to some extent
even in the contexts where mechanism design is most fruitfully applied.
For example, one can imagine that it would be difficult and costly for a
major search engine to entirely redesign its existing auction mechanism for
allocating advertisement space, because of existing users' expectations,
interfacing software, etc.  But this does not mean that aspects of the game
played by the advertisers cannot be tweaked in the designer's favor.

In this paper, we introduce a general framework for addressing intermediate
cases, where the designer has some but not full control over the game.  We
focus on {\em incompletely specified games}, where some entries of the game
matrix contain {\em sets} of payoffs, from among which the designer must
choose.  The designer's aim is to choose so that the resulting equilibrium
of the game is desirable to her.
This problem is conceptually related to $k$-implementation~\citep{Monderer04:K} and the closely related internal
implementation~\citep{Anderson10:Internal}, where one of the parties is
also able to modify an existing game to achieve better equilibria for
herself. However, in those papers the game is modified by committing to
payments, whereas we focus on choosing from a fixed set of payoffs in an entry.

We focus on two-player zero-sum games, both symmetric and not (necessarily) symmetric, and show NP-hardness in both cases. (Due to a technical reason explained later, hardness for the symmetric case does not imply hardness for the not-necessarily-symmetric case.) The hardness result for the symmetric case holds even for {\em weak tournament games}, in which the payoffs are all $-1$, $0$, or $1$, and for {\em tournament games}, in which the off-diagonal payoffs are all $-1$ or $1$.

These results have direct implications for related problem in {\em computational social choice}, another important research area in multiagent systems. In social choice (specifically, voting), we take as input a vector of rankings of the alternatives (e.g.,\xspace $a \succ c \succ b$) and as output return some subset of the alternatives. Some social choice functions are based on the {\em pairwise majority graph} which has a directed edge from one alternative to another if a majority of voters prefers the former.
One attractive concept is that of the {\em essential set}~\citep{LLL93b,DuLa99a}, which can be thought of as based on the following weak tournament game. Two abstract players simultaneously pick an alternative, and if one player's chosen alternative has an edge to the other's, the former wins. Then, the essential set ($\es$) consists of all alternatives that are played with positive probability in some equilibrium. In the absence of majority ties, this game is a tournament game and its essential set is referred to as the {\em bipartisan set} ($\bp$).

An important computational problem in social choice is the {\em possible (necessary) winner problem}~\citep{Conitzer02:Elicitation,KoLa05a,LPR+11a,XiCo11a,ABF+15a}: given only {\em partial} information about the voters' preferences---for example, because we have yet to elicit the preferences of some of the voters---is a given alternative potentially (necessarily) one of the chosen ones?
It can thus be seen that our hardness results for incompletely specified (weak) tournament games directly imply hardness for the possible/necessary winner problems for $\es$ and $\bp$.

We conclude the paper by formulating and evaluating the efficacy of a
mixed-integer linear programming formulation for the possible equilibrium action problem in weak tournament games.
Due to the space constraint, most (details of) proofs have been omitted and can be found in the full version of this paper.

\section{Examples}

The following is an incompletely specified two-player symmetric zero-sum game with actions $a,b,c,d$.

\newcommand{\WidestEntry}{$\set{-1,0,1}$}%
\newcommand{\SetToWidest}[1]{\makebox[\widthof{\WidestEntry}]{#1}}%

\vspace{-.3em}
\begin{center}
	\renewcommand{\arraystretch}{1.2}
	  $\begin{array}{r|cccc|}
	\multicolumn{1}{c}{}&\multicolumn{1}{c}{a}&\multicolumn{1}{c}{b}&\multicolumn{1}{c}{c}&\multicolumn{1}{c}{d}\\
	    \cline{2-5}
		a & 0 & \SetToWidest{1} & 0 &  \set{-1,0,1} \\
	    b & -1 & 0 & 1 & 0  \\
	    c &  0 & -1 & 0 & 1 	\\
		d & \set{-1,0,1} & 0 & -1 & 0 \\
		\cline{2-5}
	  \end{array}$
\end{center}
\vspace{.25em}
Here, each entry specifies the payoff to the row player (since the game is
zero-sum, the column player's payoff is implicit), and the set notation
indicates that the payoff in an entry is not yet fully specified. E.g., $\{-1,0,1\}$ indicates that the designer may choose either $-1$, $0$, or
$1$ for this entry.
In the case of symmetric games, we require that the designer keep the game
symmetric, so that if she sets\footnote{Let $u_r(x,y)$ denote the payoff to the row player in row $x$ and column $y$.} $u_r(d,a)=1$ then she must also
set $u_r(a,d)=-1$. Thus, our example game has three possible completions.
The goal for the designer, then, is to choose a completion in such a way
that the equilibrium of the resulting game is desirable to her. For
example, the designer may aim to have only actions~$a$ and~$c$ played with positive
probability in equilibrium. Can she set the payoffs so that this happens?
The answer is yes, because the completion with $u_r(a,d)=1$ has this property.
Indeed, for any $p \ge \frac{1}{2}$, the mixed strategy $pa+(1-p)c$ is an equilibrium strategy for this completion (and no other equilibrium strategies exist).
On the other hand, the completion with $u_r(a,d)=-1$ does have Nash equilibria in which $b$ and $d$ are played with positive probability (for example, both players mixing uniformly is a Nash equilibrium of this game).

Next, consider the following incompletely specified asymmetric zero-sum game:
\vspace{-.5em}
\begin{center}
	\renewcommand{\arraystretch}{1.2}
	\renewcommand{\WidestEntry}{$\set{-1,1}$}%
	  $\begin{array}{r|cc|}
	\multicolumn{1}{c}{}&\multicolumn{1}{c}{\ell}&\multicolumn{1}{c}{r}\\
	    \cline{2-3}
  	  t & -2 & \SetToWidest{1} \\
  	  b & \set{-1,1} & 0\\
		\cline{2-3}
	  \end{array}$
\end{center}
\vspace{.25em}
Suppose the designer's goal is to \emph{avoid} row $t$ being played in
equilibrium.  One might think that the best way to achieve this is to make
row $b$ (the only other row) look as good as possible, and thus set
$u_r(b,\ell)=1$. This results in a fully mixed equilibrium where~$t$ is
played with probability~$\frac{1}{4}$ (and~$\ell$ with~$\frac{1}{4}$).  On
the other hand, setting $u_r(b,\ell)=-1$ results in $\ell$ being a strictly
dominant strategy for the column player, and thus the row player would actually play
$b$ with probability $1$.

\section{Preliminaries}
\label{sec:prelims}

In this section, we formally introduce the concepts
and computational problems studied in the paper.
For a natural number $n$, let $[n]$ denote the set $\set{1,\ldots,n}$.

\subsection{Games}

A matrix $M \in \mathbb{Q}^{m \times n}$ defines a two-player zero-sum game
(or \emph{matrix game}) as follows. Let the rows of $M$ be indexed by
$I=[m]$ and the columns of $M$ be indexed by $J=[n]$, so that $M =
(m(i,j))_{i \in I, j \in J}$. Player $1$, the \emph{row player}, has action
set $I$ and player $2$, the \emph{column player}, has action set~$J$. If
the row player plays action $i \in I$ and the column player plays action $j
\in J$, the payoff to the row player is given by $m(i,j)$ and the payoff to
the column player is given by $-m(i,j)$. A \emph{(mixed) strategy} of the
row (resp., column) player is a probability distribution over $I$ (resp.,
$J$).
Payoffs are
extended to mixed strategy profiles in the usual way.

A matrix game $M=(m(i,j))_{i \in I, j \in J}$ is \emph{symmetric} if $I=J$
and $m(i,j)= -m(j,i)$ for all $(i,j) \in I \times J$.
A \emph{weak tournament game} is a symmetric matrix game in which all
payoffs are from the set $\set{-1,0,1}$.  Weak tournament games naturally
correspond to directed graphs $W=(A,\succ)$ as follows: vertices
correspond to actions and there is a directed edge from
action $a$ to action $b$ (denoted $a \succ b$) if and only if the payoff to the row player in
action profile $(a,b)$ is~$1$.
A \emph{tournament game} is a weak tournament game with the additional property that the payoff is $0$ \emph{only if} both players choose the same action. The corresponding graph thus has a directed edge for every pair of (distinct) vertices.

\subsection{Incomplete Games}

An \emph{incompletely specified matrix game} (short: \emph{incomplete
  matrix game}) is given by a matrix $M \in (2^\mathbb{Q})^{m \times
  n}$. That is, every entry of the matrix $M = (m(i,j))_{i \in I, j \in J}$
is a subset $m(i,j) \subseteq \mathbb{Q}$. If $m(i,j)$ consists of a single
element,
we say that the payoff for action profile $(i,j)$ is \emph{specified}, and
write $m(i,j)=m$ instead of the more cumbersome $m(i,j)=\set{m}$.
For an incomplete matrix game, the set of \emph{completions} is given by the
set of all matrix games that arise from selecting a number from the corresponding set for every action profile for which the payoff is unspecified.

An \emph{incomplete symmetric game} is an incomplete matrix game with $m(j,i) =
\set{-m: m \in m(i,j)}$ for all $i \in I$ and $j \in J$. The set of
\emph{symmetric completions} of an incomplete symmetric game is given
by the set of all completions that are symmetric.  When considering
incomplete symmetric games, we will restrict attention to symmetric
completions, which is the reason hardness results do not transfer from the
symmetric case to the general case.
An \emph{incomplete weak tournament game} is an incomplete symmetric game for which
(1) every unspecified payoff has the form $m(i,j)=\set{-1,0,1}$ with $i \neq j$, and
(2) every symmetric completion is a weak tournament game.
An \emph{incomplete tournament game} is an incomplete symmetric game for which
(1) every unspecified payoff has the form $m(i,j)=\set{-1,1}$ with $i \neq j$, and
(2) every symmetric completion is a tournament game.
Every incomplete (weak) tournament game corresponds to a directed graph in which the relation for certain pairs $(i,j)$ of distinct vertices is unspecified. Whereas every completion of an incomplete tournament game satisfies either $m(i,j)=1$ or $m(i,j)=-1$ for any such pair, a completion of an incomplete \emph{weak} tournament game also allows ``ties,'' i.e.,\xspace $m(i,j)=0$.

\subsection{Equilibrium Concepts}

The standard solution concept for normal-form games is Nash equilibrium. A strategy profile $(\sigma,\tau)$ is a Nash equilibrium of a matrix game $M$ if the strategies $\sigma$ and $\tau$ are best responses to each other, i.e.,\xspace $m(\sigma,j) \ge m(\sigma,\tau) \ge m(i,\tau)$ for all $i \in I$ and $j \in J$. The payoff to the row player is identical in all Nash equilibria, and is known as the \emph{value} of the game.

We are interested in the question whether an action is played with positive
probability in at least one Nash equilibrium.
For improved readability, the following definitions are only formulated for the row player; definitions for the column player are analogous.
The support $\supp(\sigma)$ of a strategy $\sigma$ is the set of actions that are played with positive probability in $\sigma$.
An action $i \in I$ is called \emph{essential}
if there exists a Nash equilibrium $(\sigma,t)$ with $i \in \supp(\sigma)$. By $\es_\text{row}(M)$ we denote the set of all actions $i \in I$ that are essential.

\begin{definition}
	The \emph{essential set} $\es(M)$ of a matrix game~$M$ contains all actions that are essential, i.e.,\xspace $\es(M) = \es_\text{row}(M) \cup \es_\text{column}(M)$.
\end{definition}

There is a useful connection between the essential set and \emph{quasi-strict} (Nash) equilibria. Quasi-strictness is a refinement of Nash equilibrium that requires that every best response is played with positive probability \citep{Hars73a}. Formally, a Nash equilibrium $(\sigma,t)$ of a matrix game $M$ is \emph{quasi-strict} if $m(\sigma,j) > m(\sigma,\tau) > m(i,\tau)$ for all $i \in I \setminus \supp(\sigma)$ and $j \in J \setminus \supp(\tau)$.
Since the set of Nash equilibria of a matrix game $M$ is convex, there
always exists a Nash equilibrium $(\sigma,\tau)$ with $\supp(\sigma) \cup
\supp(\tau) = \es(M)$. Moreover, it has been shown that all quasi-strict
equilibria of a matrix game have the same support \citep{BrFi08a}.
Thus, an action is contained in the essential set of a matrix game if and only if it is played with positive probability in some quasi-strict Nash equilibrium.
\citet{BrFi08a} have shown that quasi-strict equilibria, and thus the essential set, can be computed in polynomial time.

\subsection{Computational Problems}
\label{sec:comp}

\newcommand{Equilibrium Containment Problem\xspace}{Equilibrium Containment Problem\xspace}
\newcommand{ECP\xspace}{ECP\xspace}

We are interested in the computational complexity of the following decision problems.
\begin{itemize}
	\item \textbf{Possible Equilibrium Action:}
Given an incomplete matrix game $M$ and an action $a$, is there a completion $M'$ of $M$ such that $a \in \es(M')$?
	\item \textbf{Necessary Equilibrium Action:}
Given an incomplete matrix game $M$ and an action $a$, is it the case that $a \in \es(M')$ for all completions $M'$ of $M$?
\end{itemize}
One may wonder why these are the right problems to solve.
Most generally, the designer could have a utility for each possible outcome (i.e.,\xspace action profile) of the game.  The next proposition shows that hardness of the possible
equilibrium action problem immediately implies hardness of the problem of maximizing
the designer's utility.
\begin{proposition}
  Suppose
the possible equilibrium action problem is NP-hard.
Then, if the designer's payoffs are nonnegative, no
  positive approximation guarantee for the designer's utility (in the
  optimistic model where the best equilibrium for the designer is chosen) can be given
  in polynomial time unless P=NP.
\end{proposition}
\begin{proof}
  Suppose, for the sake of contradiction, that there is a polynomial time
  algorithm with a positive approximation guarantee for the problem of
  maximizing the designer's optimistic utility.  Then we can use this
  algorithm for determining whether there is a completion where a strategy
  receives positive probability in some equilibrium: simply give the
  designer utility $1$ for all outcomes in which that strategy is played,
  and $0$ everywhere else.  The designer can get the strategy to be played
  with positive probability if and only if she can obtain positive utility
  from this game, and she can obtain positive utility from this game if and
  only if the approximation algorithm returns a positive utility.
\end{proof}
A similar connection can be given between the necessary equilibrium action
problem and the case where designer utilities are
nonpositive and a pessimistic model is used (assigning a payoff of $-1$
to the action in question and $0$ otherwise).

In the context of weak tournament games, the essential set ($\es$) is often interpreted as a (social) choice function identifying desirable alternatives \citep{DuLa99a}. In the special case of tournament games, the essential set is referred to as the \emph{bipartisan set} ($\bp$) \citep{LLL93b}.
The possible and necessary equilibrium action problems defined above thus correspond to possible and necessary \emph{winner} queries for the social choice functions $\es$ (for weak tournament games) and $\bp$ (for tournament games).
The computational complexity of possible and necessary winners has been studied for many common social choice functions (e.g.,\xspace \citealp{XiCo11a}; \citealp{ABF+15a}). To the best of our knowledge, we are the first to provide complexity results for $\es$ and $\bp$.

\section{Zero-Sum Games}
\label{sec:zerosum}

\newcommand{alternating\xspace}{alternating\xspace}
\newcommand{\mathit{tr}}{\mathit{tr}}

In this section, we show that computing possible and necessary equilibrium actions is intractable for (not-necessarily-symmetric) matrix games. In the proofs, we will make use of a class of games that we call \emph{alternating\xspace games}. Intuitively, an alternating\xspace
game is a generalized version of Rock-Paper-Scissors that additionally
allows ``tiebreaking payoffs'' which are small payoffs in cases where both
players play the same action. A formal definition and proofs of some required properties are given in the full version of this paper.

We first consider the \emph{necessary} equilibrium action problem.
For ease of readability, we only give an informal proof sketch here. Much of the work in the complete proof (to be found in the appendix) is to correctly set
values for constants so that the desired equilibrium properties hold.

\begin{theorem} \label{thm:matrix-nec}
	The necessary equilibrium action problem (in matrix games that are not
        necessarily symmetric) is {coNP-complete}.
\end{theorem}

\begin{proof}[Proof sketch]

	For NP-hardness, we give a reduction from \textsc{SetCover}. An instance of \textsc{SetCover} is given by a collection $\{ S_1, \ldots, S_n \}$ of subsets of a universe $U$, and an integer~$k$; the question is whether we can cover $U$ using only $k$ of the subsets.
We may assume that $k$ is odd (it is always possible to add a singleton
subset with an element not covered by anything else and increase $k$ by $1$). Define an incomplete matrix game $M$ where the row player has $3n-k+1$ actions, and the column player has $2n-k+|U|$ actions. The row player's actions are given by $\{ S_{i,j} : i \in [n], j \in [2] \} \cup \{ x_i : i \in [n-k]\} \cup \{ r_* \}$.

		\begin{figure}
		\renewcommand{\arraystretch}{1.8}
		\begin{center}
\footnotesize
\setlength{\tabcolsep}{0.19em}
\scalebox{1.2}{
\begin{tabular}{r|c|c|c|c|c||c|c|c|c|c|}

		    		  \multicolumn{1}{c}{} & \multicolumn{1}{c}{$c_1$} & \multicolumn{1}{c}{$c_2$} & \multicolumn{1}{c}{$c_3$} & \multicolumn{1}{c}{$c_4$} & \multicolumn{1}{c}{$c_5$} & \multicolumn{1}{c}{$s_1$} & \multicolumn{1}{c}{$s_2$} & \multicolumn{1}{c}{$s_3$} & \multicolumn{1}{c}{$s_4$} & \multicolumn{1}{c}{$s_5$}  \\ \cline{2-11}
		    	    $S_{1,1}$ & $0$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & \phantom{h} $\scalebox{0.75}[1.0]{\( - \)}H$ \phantom{h} & $y$ & $y$ & $0$ & $0$ & $0$  \\ \cline{2-11}
		    	    $S_{1,2}$ & $\{\scalebox{0.75}[1.0]{\( - \)}1, 1 \}$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $x$ & $x$ & $0$ & $0$ & $0$  \\  \cline{2-11}
		    	    $S_{2,1}$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $0$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $0$ & $0$ & $y$ & $y$ & $0$ \\          \cline{2-11}
		    	    $S_{2,2}$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\{\scalebox{0.75}[1.0]{\( - \)}1, 1 \}$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $0$ & $0$ & $x$ & $x$ & $0$\\ \cline{2-11}
		    	    $S_{3,1}$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $0$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $0$ & $0$ & $y$ & $0$ & $y$\\          \cline{2-11}
		    	    $S_{3,2}$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\{\scalebox{0.75}[1.0]{\( - \)}1, 1 \}$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $0$ & $0$ & $x$ & $0$ & $x$ \\ \cline{2-11}
		    	    $S_{4,1}$ & ${\scalebox{0.75}[1.0]{\( - \)}}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $0$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$  & $0$ & $y$ & $0$ & $y$ & 0 \\          \cline{2-11}
		    	    $S_{4,2}$ & ${\scalebox{0.75}[1.0]{\( - \)}}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\{\scalebox{0.75}[1.0]{\( - \)}1, 1 \}$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ &  $0$ & $x$ & $0$ & $x$ & $0$\\ \cline{2-11}
		    		$x_1$     & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\phantom{\scalebox{0.75}[1.0]{\( - \)}}H$ & $\scalebox{0.75}[1.0]{\( - \)}1$ & $0$ & $0$ & $0$ & $0$ & $0$  \\ \hhline{~|=|=|=|=|=#-|-|-|-|-|}
\multicolumn{1}{c|}{$r_*$} & \multicolumn{1}{c|}{$-v$} & \multicolumn{1}{c|}{$-v$} & \multicolumn{1}{c|}{$-v$} & \multicolumn{1}{c|}{$-v$} & \multicolumn{1}{c|}{$-v$} & \multicolumn{1}{c|}{G} & \multicolumn{1}{c|}{G} & \multicolumn{1}{c|}{G} & \multicolumn{1}{c|}{G} & \multicolumn{1}{c|}{G}

\\       \cline{2-11}
		  \end{tabular}
		  }
		\caption{The incomplete matrix game $M$ used in the proof of \thmref{thm:matrix-nec} for the \textsc{SetCover} instance given by {$|U|=5$}, $n=4$, $k=3$, $S_1=\{s_1, s_2 \}$, $S_2 = \{s_3, s_4 \}$, $S_3=\{s_3, s_5 \}$, and $S_4 = \{ s_2, s_4 \}$. $L$ lies in the top left, indicated by double lines.
		}
		\end{center}
		\end{figure}

	            Let $L$ denote the restriction of the game to the first
                $2n-k$ columns and $3n-k$ rows. We denote the column player's actions in
                this part of the game by $c_1, \ldots, c_{2n-k}$. We
                set $m(S_{i,1},c_i)=0$ and $m(S_{i,2},c_i)=\{ -1, 1 \}$ for all $i \in [n]$ and
                $m(x_{i}, c_{n+i})=-1$ for all $i \in [n-k]$. We fill in the remaining entries with
                $H$ and $-H$, where $H$ is a large positive number, so that if we consider only one of each pair of rows $\{ S_{i,1}$, $S_{i,2} \}$, $L$ acts as an alternating\xspace game. Setting $m(S_{i,2},c_i)=-1$ will correspond to choosing $S_i$ for the set cover, and setting $m(S_{i,2},c_i)=1$ will correspond to not choosing $S_i$. Note that, considering only $L$, the row player will put positive probability on exactly one of $S_{i,1}$ and $S_{i,2}$ (as well as all rows $x_i$) and, as long as $H$ is sufficiently large, each row that is played with positive probability receives approximately $\frac{1}{N}$ probability. $S_{i,1}$ is played if $S_i$ is chosen for the set cover, $S_{i,2}$ is played otherwise. Also note that the value of $L$ is close to zero, depending on the exact setting of the undetermined entries.

We have additional columns $s_1, \ldots, s_{|U|}$ corresponding to elements of $U$. For every set $S_j$ containing $s_i$, column $s_i$ has a positive entry $y$ in row $S_{j,1}$ and a negative entry $x$ in row $S_{j,2}$.
If $s_i$ is not covered by any chosen set, then the equilibrium can not be contained in $L$: If it were, then the column player could best respond by playing $s_i$, where all entries (on rows played with positive probability by the row player) are either $0$ or $x<0$. However, if $s_i$ is covered by some set, then we can make $y$ large enough (relative to~$x$) that the column player will not play $s_i$. Thus, if every $s_i$ is covered, the column player plays only columns from $L$.

Finally, we have a single extra row labeled $r_*$. This row has a small negative payoff $-v$ for all columns in $L$, and a very large positive payoff $G$ for all columns not in $L$. As long as the equilibrium is contained in $L$ (i.e.,\xspace all elements are covered), it is not a best response for the row player to play~$r_*$. However, if the column player puts positive probability on some $s_i$ (that is, $s_i$ is uncovered), then $G$ is large enough that the row player can best respond by playing $r_*$ with positive probability.
\end{proof}

By modifying the construction in the proof of \thmref{thm:matrix-nec}, we also get a hardness result for the problem of deciding whether an action is a \emph{possible} equilibrium action.

\begin{theorem} \label{thm:matrix-pos}
	The possible equilibrium action problem (in matrix games that are not
        necessarily symmetric) is NP-complete.
\end{theorem}

\section{Weak Tournament Games}
\label{sec:weak}

We now turn to weak tournament games and analyze the computational complexity of possible and necessary $\es$ winners.

\begin{theorem} \label{thm:weak-pos}
	The possible $\es$ winner problem (in weak tournament games) is NP-complete.
\end{theorem}

\begin{proof}[Proof sketch]

	For NP-hardness, we provide a reduction from \textsc{Sat}.
	 %
	Let $\varphi = C_1 \land \ldots \land C_m$ be a Boolean formula in conjunctive normal form over a finite set $V= \{v_1, \ldots, v_n\}$ of variables.
	We define an incomplete weak tournament\footnote{We utilize the one-to-one correspondence between weak tournament games and directed graphs without cycles of length one or two (so-called \emph{weak tournaments}). For a weak tournament $(A,\succ)$, we use the notation $a \succ b$ to denote a directed edge from~$a$~to~$b$.} $W_\varphi=(A,\succ)$ as follows.
	The set $A$ of vertices is given by $A =  \cup_{i=1}^n X_i \cup \set{c_1, \ldots, c_m} \cup \set{d}$, where $X_i = \set{x_i^1,\ldots,x_i^6}$ for all $i \in [n]$. Vertex $c_j$ corresponds to clause $C_j$ and the set $X_i$ corresponds to variable~$v_i$.

	Within each set $X_i$, there is a cycle
$x_i^1 \succ x_i^2 \succ x_i^3 \succ x_i^4 \succ x_i^5 \succ x_i^6 \succ x_i^1$ and an unspecified edge between $x_i^1$ and $x_i^4$.
	If variable $v_i$ occurs as a positive literal in clause $C_j$, we
        have edges $c_j \succ x_i^3$ and $x_i^5 \succ c_j$. If variable
        $v_i$ occurs as a negative literal in clause $C_j$, we have edges
        $c_j \succ x_i^6$ and $x_i^2 \succ c_j$. Moreover, there is an edge from $c_j$ to $d$ for every $j \in [m]$. For all pairs of vertices for which neither an edge has been defined, nor an unspecified edge declared, we have a tie.  See \figref{fig:construction} for an example.

	We make two observations about $W_\varphi$.

	\begin{observation} \label{obs1}
		For each completion $W$ of $W_\varphi$, we have $d \in \es(W)$ if and only if $\es(W) \cap \set{c_1, \ldots, c_m} = \emptyset$.
	\end{observation}

	\begin{observation} \label{obs2}
		For each $i$, there is exactly one unspecified edge within (and thus exactly three possible completions of) the subtournament $W_\varphi|_{X_i}$. If the we set a tie between $x_i^1$ and $x_i^4$, then all Nash equilibria $p$ of the subtournament $W_\varphi|_{X_i}$ satisfy $p(x_i^1)=p(x_i^3)=p(x_i^5)$ and $p(x_i^2)=p(x_i^4)=p(x_i^6)$. If we set $x_i^1 \succ x_i^4$, then every quasi-strict equilibrium $p$ of $W_\varphi|_{X_i}$ satisfies $p(x_i^2)=p(x_i^4)=p(x_i^6)=0$, $p(x_i^5)>p(x_i^1)>p(x_i^3)>0$, and $p(x_i^1)+p(x_i^3)>p(x_i^5)$.
	By symmetry, setting $x_i^4 \succ x_i^1$ results in quasi-strict equilibria $p$ with $p(x_i^1)=p(x_i^3)=p(x_i^5)=0$, $p(x_i^4)>p(x_i^6)>p(x_i^2)>0$, and $p(x_i^2)+p(x_i^6)>p(x_i^4)$.
	\end{observation}

	We can now show hat $\varphi$ is satisfiable if and only if there is a completion $W$ of $W_\varphi$ with $d \in \es(W)$. For the direction from left to right, let $\alpha$ be a satisfying assignment and consider the completion $W$ of $W_\varphi$ as follows: if $v_i$ is set to ``true'' under $\alpha$, add edge $x_i^1 \succ x_i^4$; otherwise, add edge $x_i^4 \succ x_i^1$.
	It can be shown that $\es(W) = \cup_{i \in [n]} \es(W|_{X_i}) \cup \set{d}$.

	For the direction from right to left, let $W$ be a completion of $W_\varphi$ with $d \in \es(W)$. Define the assignment $\alpha$ by setting variable $v_i$ to ``true'' if $x_i^1 \succ x_i^4$ and to ``false'' if $x_i^4 \succ x_i^1$. If there is a tie between $x_i^1$ and $x_i^4$, we set the truth value of $v_i$ arbitrarily.
	Since $d \in ES(W)$, we know by \obsref{obs1} that $c_j \notin \es(W)$ for all $j \in [m]$. It can now be shown that every $c_i$ has an incoming edge from a vertex in $\es(W)$, and that this vertex corresponds to a literal that appears in $C_i$ and that is set to ``true'' under $\alpha$.
\end{proof}

\begin{figure}[tb]
\centering
\begin{tikzpicture}[scale=1.2]
	\tikzstyle{every node}=[circle,draw,minimum size=1.4em,inner sep=0pt]

	\draw(-4,3.5) node(d){$d$};

	\draw (0,3.5) node(c1){$c_1$} ++(1.5,0) node(c2){$c_2$} ++(1.5,0) node(c3){$c_3$};

		\draw[-latex] (c1) -- (d);
		\draw[-latex] (c2) .. controls  (0,4) and (-2.25,4) .. (d);
		\draw[-latex] (c3) .. controls  (0,4.5) and (-2.5,4.5) .. (d);

	\draw (-3,2) node(x11){$x_1^1$}
	++(0.5,-.75) node(x12){$x_1^2$}
	++(0.5,-.75) node(x13){$x_1^3$}
	++(-1,0)     node(x14){$x_1^4$}
	++(-1,0)     node(x15){$x_1^5$}
	++(0.5,.75)  node(x16){$x_1^6$};
	\foreach \x / \y in {x11/x12,x12/x13,x13/x14,x14/x15,x15/x16,x16/x11}
		{ \draw[-latex,] (\x) -- (\y); }
	\draw[dashed] (x11) -- (x14);

	\draw (0,2) node(x21){$x_2^1$}
	++(0.5,-.75) node(x22){$x_2^2$}
	++(0.5,-.75) node(x23){$x_2^3$}
	++(-1,0)     node(x24){$x_2^4$}
	++(-1,0)     node(x25){$x_2^5$}
	++(0.5,.75)  node(x26){$x_2^6$};
	\foreach \x / \y in {x21/x22,x22/x23,x23/x24,x24/x25,x25/x26,x26/x21}
		{ \draw[-latex,] (\x) -- (\y); }
	\draw[dashed] (x21) -- (x24);

	\draw (3,2) node(x31){$x_3^1$}
	++(0.5,-.75) node(x32){$x_3^2$}
	++(0.5,-.75) node(x33){$x_3^3$}
	++(-1,0)     node(x34){$x_3^4$}
	++(-1,0)     node(x35){$x_3^5$}
	++(0.5,.75)  node(x36){$x_3^6$};
	\foreach \x / \y in {x31/x32,x32/x33,x33/x34,x34/x35,x35/x36,x36/x31}
		{ \draw[-latex,] (\x) -- (\y); }
	\draw[dashed] (x31) -- (x34);

	\draw[-latex] (c1) .. controls (-2,2.5)  .. (x13);
	\draw[-latex] (x15) .. controls (-4,2.5)  .. (c1);

	\draw[-latex] (c1) to [bend right=15] (x26);
	\draw[-latex] (x22) to [bend right=15] (c1);

	\draw[-latex] (c1) .. controls (4,2.5)  .. (x33);
	\draw[-latex] (x35).. controls (2,2.5)  .. (c1);

\end{tikzpicture}
\caption{The weak tournament $W_\varphi$ for formula $\varphi= C_1 \wedge C_2 \wedge C_3$ with $C_1 = x_1 \lor \lnot x_2 \lor x_3$. Dashed lines indicate unspecified edges. For improved readability, edges connecting $c_2$ and $c_3$ to $X$ have been omitted.}
\label{fig:construction}
\end{figure}

We get hardness for the necessary winner problem by slightly modifying the construction used in the proof above.

\begin{theorem} \label{thm:weak-nec}
	The necessary $\es$ winner problem (in weak tournament games) is coNP-complete.
\end{theorem}

It can actually be shown that the problems considered in Theorems \ref{thm:weak-pos} and \ref{thm:weak-nec} remain intractable even in the case where unspecified payoffs can be chosen from the interval $[-1,1]$ (while still maintaining symmetry). This is interesting insofar as this is our only hardness result for \emph{infinite and/or continuous} payoff sets; such relaxations often make problems computationally easier.

\begin{proposition} \label{prop:continuous}
	The possible equilibrium action problem in weak tournament games remains NP-complete (and the necessary equilibrium action problem coNP-complete) when every payoff set $m(i,j)$ with $m(i,j)=\set{-1,0,1}$ is replaced by $m(i,j)=[-1,1]$.
\end{proposition}

\section{Tournament Games}
\label{sec:tournaments}

The hardness results in \secref{sec:weak} leave open the possibility that computing $\es$ is tractable in tournament games (where $\es$ is referred to as $\bp$). Indeed, it often turns out that computational problems become easier to solve when restricting attention to tournaments \citep{BBFHR11a,BrFi08b,KMSc07a}.\footnote{There are also cases in the literature where a computational problem remains hard when restricted to tournaments, but the hardness proof is much more complicated \citep{Alon06a,CTY07a,Coni06a}.}
The reason is that certain structural properties only hold in tournaments.\footnote{For example, \citet{LLL93a} and \citet{FiRy92a} have shown that every tournament game $T$ has a \emph{unique} Nash equilibrium. This Nash equilibrium is quasi-strict and has support $\es(T)=\bp(T)$.}

Nevertheless, we prove that computing possible and necessary $\es$ winners is hard even in tournament games. The technical difficulty in proving these results lies in the fact that the hardness reduction cannot use ``ties'' (i.e.,\xspace non-edges) in (the specified part of) the graph.

\begin{theorem} \label{thm:posBP}
	The possible $\bp$ winner problem (in tournament games) is NP-complete.
\end{theorem}

\begin{theorem} \label{thm:necBP}
	The necessary $\bp$ winner problem (in tournament games) is coNP-complete.
\end{theorem}

Observe that these results neither imply the results for weak tournament games in \secref{sec:weak} (where completions can use ``ties'') nor the results for general matrix games in \secref{sec:zerosum} (where completions can be asymmetric).

\section{MIP for Weak Tournament Games}
\label{sec:mip}

Of course, the fact that a problem is NP-hard does not make it go away; it
is still desirable to find algorithms that scale reasonably well (or very
well on natural instances).  NP-hard problems in game theory often allow
such algorithms.  In particular, formulating the problem as a \emph{mixed-integer
program (MIP)} and calling a general-purpose solver often provides good results.
In this section, we formulate the possible $\es$ winner problem in weak tournament games as a MIP.

\subsection{Mixed-Integer Programming Formulation}

Let $W=(w(i,j))_{i,j \in A}$ be an incomplete weak tournament game.
For every entry $w(i,j)$ of $W$, we define two binary variables $x_{ij}^{\text{pos}}$ and $x_{ij}^{\text{neg}}$.
Setting $w(i,j)$ to $w_{ij} \in \set{-1,0,1}$ corresponds to setting $x_{ij}^{\text{pos}}$ and $x_{ij}^{\text{neg}}$ in such a way that $(x_{ij}^{\text{pos}}, x_{ij}^{\text{neg}}) \neq (1,1)$ and $x_{ij}^{\text{pos}} - x_{ij}^{\text{neg}} = w_{ij}$.
For each action $j$, there is a variable $p_j$ corresponding to the probability that the column player assigns to $j$.
Finally, $z_{ij}$ is a variable that, in every feasible solution, equals $w_{ij} p_j$.

To determine whether an action $k \in A$ is a possible $\es$ winner of $W$, we solve the following MIP. Every feasible solution of this MIP corresponds to a completion of $W$ and a Nash equilibrium of this completion.
  \begin{alignat*}{3}
	  &\text{maximize } p_k && \\
	  &\text{subject to} && \\
	   	&x_{ij}^{\text{neg}}-x_{ji}^{\text{pos}}=0, \, \forall i,j 	& \quad 		   		&x_{ij}^{\text{pos}}=1, \, \text{if } w(i,j)=1\\
		&x_{ij}^{\text{pos}} + x_{ij}^{\text{neg}} \le 1, \,  \forall i,j & \quad 			&x_{ij}^{\text{neg}}=1, \, \text{if } w(i,j)=-1 \\
		&x_{ij}^{\text{pos}}=x_{ij}^{\text{neg}}=0, \, \text{if } w(i,j)=0 & \quad 			&x_{ij}^{\text{pos}}, x_{ij}^{\text{neg}} \in \set{0,1}, \, \forall i,j\\
		&z_{ij} \ge p_j-2(1-x_{ij}^{\text{pos}}), \, \forall i,j & 			   				&\textstyle{\sum_{j \in A} z_{ij}} \le 0,  \, \forall i \\
		&z_{ij} \ge -p_j-2(1-x_{ij}^{\text{neg}}), \, \forall i,j & 			   			&\textstyle{\sum_{j \in A}p_j} = 1\\
		&z_{ij} \ge -2x_{ij}^{\text{pos}} - 2x_{ij}^{\text{neg}}, \, \forall i,j &			&p_j \ge 0, \, \forall j\\
   \end{alignat*}

\vspace{-0.5cm}
  Here, indices $i$ and $j$ range over the set $A$ of actions.
  Most interesting are the constraints on $z_{ij}$; we note that exactly
  one of the three will be binding depending on the values of
  $x_{ij}^{\text{pos}}$ and $x_{ij}^{\text{neg}}$. The net effect of these constraints is
  to ensure that $z_{ij} \ge w_{ij}p_j$. (Since we also have the constraint $\sum_{j \in A} z_{ij} \le 0$ and because the value of every completion is zero, $z_{ij} = w_{ij}p_j$ in every feasible solution.)
  All other constraints containing $x_{ij}^{\text{pos}}$ or $x_{ij}^{\text{neg}}$ are to impose symmetry and consistency on the entries. The remaining constraints make sure that $p$ is a well-defined probability distribution and that no row yields positive payoff for player $1$.

  It is possible to adapt this MIP to compute possible and necessary $\bp$ winners in tournament games. All that is required is to replace inequality constraints of the form $x_{ij}^{\text{pos}} + x_{ij}^{\text{neg}} \le 1$ by equalities, thus eliminating the possibility to set $w_{ij}=0$. Since tournament games have a unique equilibrium, checking whether action $k$ is a possible or necessary $\bp$ winner can be done by maximizing and minimizing the objective function $p_k$, respectively. The reason that this approach does not extend to the computation of necessary winners in \emph{weak} tournament games is that weak tournaments may have multiple equilibria, some of them not quasi-strict. Since our MIP optimizes over the set of all (not necessarily quasi-strict) equilibria, we may encounter cases where the MIP finds a completion with $p_k=0$, but $k$ is still a necessary winner because it  is played with positive probability in every \emph{quasi-strict} equilibrium.

\subsection{Experimental Results}

\begin{figure}[tb]
	\centering
    \includegraphics[width=7cm]{Graph1-bigfont-cut.pdf}
	\hfill
	\includegraphics[width=7cm]{Graph2-bigfont-cut.pdf}
  \caption{Average runtime (log scale) for $\frac{n}{2}$ unspecified entries (left) and $n$ unspecified entries (right).}
  \label{fig:graphs}
\end{figure}

We tested our MIP for the possible $\es$ winner problem in weak tournament games containing either $\frac{n}{2}$ or $n$ unspecified entries, where $n=|A|$ is the number of actions available to each player. For each $n$, we examined the average time required to solve 100 random instances\footnote{Random instances were generated by randomly choosing each entry from $\{-1,0,1\}$ and imposing symmetry, then randomly choosing the fixed number of entries to be unspecified.} of size~$n$, using CPLEX 12.6 to solve the MIP.  Results are shown in \figref{fig:graphs}, with algorithms cut off once the average time to find a solution exceeds 10 seconds.

We compared the performance of our MIP with a simple brute
force algorithm. The brute force algorithm performs a  depth-first search over the space of all completions, terminating when it finds a certificate of a yes instance or after it has exhausted all completions.
We observe that for even relatively small values of $n$, the MIP begins to
significantly outperform the brute-force algorithm.

\section{Conclusion}
\label{sec:conclusion}

Often, a designer has some, but limited, control over the game being
played, and wants to exert this control to her advantage.  In this paper,
we studied how computationally hard it is for the designer to decide
whether she can choose payoffs in an incompletely specified game to achieve
some goal in equilibrium, and found that this is NP-hard even in quite
restricted cases of two-player zero-sum games.
Our framework and our results also apply in cases where there is no designer but we are just uncertain about the payoffs, either because further exploration is needed to determine what they are, or because they vary based on conditions (e.g., weather). In such settings one might simply be interested in potential and unavoidable equilibrium outcomes.

Future work may address the
following questions.  Are there classes of games for which these problems
are efficiently solvable?  Can we extend the MIP approach to broader
classes of games?  What results can we obtain for general-sum games?  Note
that just as hardness for symmetric zero-sum games does not imply hardness
for zero-sum games in general (because in the latter the game does not need
to be kept symmetric), in fact hardness for zero-sum games does not imply
hardness for general-sum games (because in the latter the game does not
need to be kept zero-sum).  However, this raises the question of which
solution concept should be used---Nash equilibrium, correlated equilibrium,
Stackelberg mixed strategies, etc.  (All of these coincide in two-player
zero-sum games.)  All in all, we believe that models where a designer has
limited, but not full, control over the game are a particularly natural
domain of study for AI researchers and computer scientists in general, due
to the problems' inherent computational complexity and potential to address
real-world settings.

\section*{Acknowledgements}

We are thankful for support from NSF under awards IIS-1527434,
IIS-0953756, CCF-1101659, and CCF-1337215, ARO under grants
W911NF-12-1-0550 and W911NF-11-1-0332, ERC under StG 639945 (ACCORD), a Guggenheim Fellowship, and a Feodor Lynen research fellowship of the Alexander von Humboldt
Foundation.
This work was done in part while Conitzer was visiting the Simons
Institute for the Theory of Computing.
We thank Martin Bullinger for helpful comments.

redpajama_set_name: RedPajamaArXiv


Lots of videos for re-teaching, practice problems and quizzes.
The websites listed below are games and teaching resources found in math class.
Use this link for exploring PARCC.
Practice problems by subject with answers worked out.
Use this link to graph any equation online.
Tutoring available every morning at 7:30 and afterschool most afternoons.
Log on to Information Now to check grades often.


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Disney released a YouTube video with concept art and some footage of "Epcot Forever," the new nighttime spectacular that will debut on October 1, 2019 – the day after "Illuminations: Reflections of Earth" closes on September 30, 2019.
Disney Cast Members in the video describe how the new show will feature music from Epcot, together with fireworks, fountains, and illuminated kites pulled by jet skis.
Excited for the "Epcot Forever" show starting October 1, 2019?