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the prisoner s dilemma ( pd ) is perhaps the most famous model in the field of game theory . roughly speaking , there are two sorts of pd : one - shot pd and iterated pd . nowadays a lot of studies on pd are focused on the latter case .
for example , axelrod @xcite investigated the evolution of cooperative behavior in well - mixed populations of selfish agents by using pd as a paradigm .
nowak and may @xcite induced spatial structure in pd , i.e. , agents were restricted to interact with his immediate neighbors .
santos and pacheco @xcite found that when agents interacted following scale - free networks , cooperation would become a dominating trait throughout the entire range of parameters of pd .
perc and szolnoki @xcite proposed that social diversity could induce cooperation as the dominating trait throughout the entire range of parameters of pd .    compared with the iterated pd
, the one - shot pd is usually viewed simple . in the original version of one - shot pd ,
two prisoners are arrested by a policeman .
each prisoner must independently choose a strategy between `` confessing '' ( denoted as strategy `` _ _ defect _ _ '' ) and `` not confessing '' ( denoted as strategy `` _ _ cooperate _ _ '' ) .
the payoff matrix of prisoners is shown in table 1 . as long as two agents are rational
, the unique nash equilibrium shall be ( _ defect _ , _ defect _ ) , which results in a pareto - inefficient payoff @xmath0 .
that is the dilemma .
_ table 1 : the payoff matrix of pd , where _ @xmath1 , and @xmath2 .
_ the first entry in the parenthesis denotes the payoff of agent 1 and the second entry stands for the payoff of agent 2_. +    [ cols="^,^,^",options="header " , ]     \1 ) @xmath3 : this strategy profile means two agents both participate the algorithmic model and submit parameters to the computer . according to ref .
@xcite , for each agent @xmath4 , his dominant parameters are @xmath5 and @xmath6 , which result in a pareto - efficient payoff @xmath7 .
+ 2 ) @xmath8 : this strategy profile means agent 1 participates the algorithmic model , but agent 2 takes back his channel and submits a message to the arbitrator directly . since agent 1 can observe agent 2 s action , in the end , both agents will take back their channels and submit messages to the arbitrator directly . obviously , the dominant message of each agent @xmath9 is @xmath10 , and the arbitrator will assign the pareto - inefficient payoff @xmath0 to agents .
+ 3 ) @xmath11 : this strategy profile is similar to the above case .
the arbitrator will assign @xmath12 to two agents .
+ 4 ) @xmath13 : this strategy profile means two agents both take back their channels and send messages to the arbitrator directly .
this case is similar to the case 2 .
the arbitrator will assign @xmath12 to two agents .    from table 3
, it can be seen that @xmath3 and @xmath13 are two nash equilibria , and the former is pareto - efficient .
as specified by telser ( page 28 , line 2 , @xcite ) , `` _ _ a party to a self - enforcing agreement calculates whether his gain from violating the agreement is greater or less than the loss of future net benefits that he would incur as a result of detection of his violation and the consequent termination of the agreement by the other party . _ _ '' since two channels have been controlled by the computer in stage 1 , in the end @xmath3 is a self - enforcing nash equilibrium and the pareto - efficient payoff @xmath7 is the unique nash equilibrium outcome . in this sense ,
the two agents escape the dilemma .    * type-5 pd * : + conditions 1 - 3 are the same as those in type-4 pd .
+ 4 ) the last condition of type-4 pd does not hold .
+ for this case , although the two agents can communicate before moving and agree that collaboration is good for each agent , they will definitely choose ( _ defect _ , _ defect _ ) as if they are separated .
thus , the agents can not escape the dilemma .
the algorithmic model revises common understanding on the one - shot pd . here
we will discuss some possible doubts about it .    _
q1 _ : the type-4 pd seems to be a cooperative game because in condition 4 , the algorithmic model constructed by two agents acts as a correlation between agents .    _
a1 _ : from the viewpoints of agents , the game is different from the original one - shot pd , since the payoff matrix confronted by the two agents has been changed from table 1 to table 3 . but from the viewpoints of the arbitrator , nothing is changed .
thus , the so - called correlation between two agents is indeed _
unobservable _ to the arbitrator .
put differently , the arbitrator can not prevent agents from constructing the algorithmic model .
+ on the other hand , since each agent can freely choose not to participate the algorithmic model and send a message to the arbitrator directly in stage 1 , the algorithmic model is self - enforcing and thus still a non - cooperative game .    _ q2 _ : after the algorithmic model is triggered , can it simply send @xmath14 , @xmath15 to the arbitrator instead of running steps 1 - 5 ?    _ a2 _ : the algorithmic model enlarges each agent s strategy space from the original strategy space _ \{cooperate , defect } _ to a two - dimensional strategy space @xmath16\times[0,\pi/2]$ ] , and generates the pareto - efficient payoff @xmath7 in nash equilibrium .
the enlarged strategy space includes the original strategy space of one - shot pd : the strategy ( _ cooperate , cooperate _ ) , ( _ cooperate , defect _ ) , ( _ defect , cooperate _ ) , ( _ defect , defect _ ) in the original pd correspond to the strategy @xmath17 , @xmath18 , @xmath19 , @xmath20 in the algorithmic model respectively , since @xmath21 , @xmath22 .
+ however , the idea in this question restricts each agent s strategy space from the original strategy space _ \{cooperate , defect } _ to a single strategy _
cooperate_. in this sense , two agents are required to sign a binding contract to do so .
this is beyond the range of non - cooperative game .
_ remark 2 _ : the algorithmic model is not suitable for type-1 and type-2 pd , because the computer can not perform actions on behalf of agents .
the algorithmic model is not applicable for type-3 pd either because two agents are separated , thereby the algorithmic model can not be constructed . for the case of type-5 pd , the algorithmic model is not applicable because condition 4 in type-4 pd is vital and indispensable .
in this paper , we categorize the well - known one - shot pd into five types and propose an algorithmic model to help two non - cooperative agents self - enforcingly escape a special type of pd , i.e. , the type-4 pd .
the type-4 pd is justified when the arbitrator communicate with the agents indirectly through some channels , and each agent s strategy is not an action , but a message that can be sent to the arbitrator . with the rapid development of internet , more and more type-4 pd games will be seen .
one point is important for the novel result : usually people may think the two payoff matrices confronted by agents and the arbitrator are the same ( i.e. , table 1 ) .
however we argue that for the case of type-4 pd , the two payoff matrices can be different : the arbitrator still faces table 1 , but the agents can self - enforcingly change their payoff matrix to table 3 by virtue of the algorithmic model , which leads to a pareto - efficient payoff .
the author is very grateful to ms .
fang chen , hanyue wu ( _ apple _ ) , hanxing wu ( _ lily _ ) and hanchen wu ( _ cindy _ ) for their great support .
the prisoner s dilemma is a simple model that captures the essential contradiction between individual rationality and global rationality . 
 although the one - shot prisoner s dilemma is usually viewed simple , in this paper we will categorize it into five different types . for the type-4 prisoner s dilemma game , we will propose a self - enforcing algorithmic model to help non - cooperative agents obtain pareto - efficient payoffs . 
 the algorithmic model is based on an algorithm using complex numbers and can work in macro applications .    
 prisoner s dilemma ; non - cooperative games .