A non-cooperative Pareto-efficient solution to the one-shot Prisoner’s Dilemma

# A non-cooperative Pareto-efficient solution to the one-shot Prisoner’s Dilemma

Haoyang Wu
###### Abstract

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.

###### keywords:
Prisoner’s Dilemma; Non-cooperative games.
Wan-Dou-Miao Research Lab, Suite 1002, 790 WuYi Road, Shanghai, 200051, China.

## 1 Introduction

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 [1] investigated the evolution of cooperative behavior in well-mixed populations of selfish agents by using PD as a paradigm. Nowak and May [2] induced spatial structure in PD, i.e., agents were restricted to interact with his immediate neighbors. Santos and Pacheco [3] 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 [4] 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 . That is the dilemma.

Table 1: The payoff matrix of PD, where , and . The first entry in the parenthesis denotes the payoff of agent 1 and the second entry stands for the payoff of agent 2.
\backslashboxagent 1agent 2 Cooperate Defect Cooperate (R, R) (S, T) Defect (T, S) (P, P)

In 1999, Eisert et al [5] proposed a quantum model of one-shot PD (denoted as EWL model). The EWL model showed “quantum advantages” as a result of a novel quantum Nash equilibrium, which help agents reach the Pareto-efficient payoff . Hence, the agents escape the dilemma. In 2002, Du et al [6] gave an experiment to carry out the EWL model.

So far, there are some criticisms on EWL model: 1) It is a new game which has new rules and thus has no implications on the original one-shot PD [7]. 2) The quantum state serves as a binding contract which let the players chooses one of the two possible moves (Cooperate or Defect) of the original game. 3) In the full three-parameter strategy space, there is no such quantum Nash equilibrium [8] [9].

Besides these criticisms, here we add another criticism: in the EWL model, the arbitrator is required to perform quantum measurements to readout the messages of agents. This requirement is unreasonable for common macro disciplines such as politics and economics, because the arbitrator should play a neutral role in the game: His reasonable actions should only receive agents’ strategies and assign payoffs to agents. Put differently, if the arbitrator is willing to work with an additional quantum equipment which helps agents to obtain the Pareto-efficient payoffs , then why does not he directly assign the Pareto-efficient payoffs to the agents?

Motivated by these criticisms, this paper aims to investigate whether a Pareto-efficient outcome can be reached by non-cooperative agents in macro applications. Note that a non-cooperative game is one in which players make decisions independently. Thus, while they may be able to cooperate, any cooperation must be self-enforcing [10].

The rest of this paper is organized as follows: in Section 2 we will propose an algorithmic model, where the arbitrator does not have to work with some additional quantum equipment (Note: here we do not aim to solve the first three criticisms on the EWL model, because these criticisms are irrelevant to the algorithmic model). In Section 3, we will categorize the one-shot PD into five different types, and claim that the agents can self-enforcingly reach the Pareto-efficient outcome for the case of type-4 PD by using the algorithmic model. The Section 4 gives some discussions. The last section draws conclusion.

## 2 An algorithmic model

As we have pointed out above, for macro applications, it is unreasonable to require the arbitrator act with some additional quantum equipment. In what follows, firstly we will amend the EWL model such that the arbitrator works in the same way as he does in classical environments, then we will propose an algorithmic version of the amended EWL model.

### 2.1 The amended EWL model

Let the set of two agents be . Following formula (4) in Ref. [9], two-parameter quantum strategies are drawn from the set:

 ^ω(θ,ϕ)≡[eiϕcos(θ/2)isin(θ/2)isin(θ/2)e−iϕcos(θ/2)],

, (where is an entanglement measure, is the Pauli matrix, is tensor product), , , .

Without loss of generality, we assume:
1) Each agent has a quantum coin (qubit), a classical card and a channel connected to the arbitrator. The basis vectors , of a quantum coin denote head up and tail up respectively.
2) Each agent independently performs a local unitary operation on his/her own quantum coin. The set of agent ’s operation is . A strategic operation chosen by agent is denoted as . If , then , ; If , then , . denotes “Not flip”, denotes “Flip”.
3) The two sides of a card are denoted as Side 0 and Side 1. The messages written on the Side 0 (or Side 1) of card is denoted as (or ). represents “Cooperate”, and represents “Defect”.
4) There is a device that can measure the state of two quantum coins and send messages to the designer.

Fig. 1 shows the amended version of EWL model (denoted as the A-EWL model). Its working steps are defined as follows:
Step 1: The state of each quantum coin is set as . The initial state of the two quantum coins is .
Step 2: Let the two quantum coins be entangled by . .
Step 3: Each agent independently performs a local unitary operation on his own quantum coin. .
Step 4: Let the two quantum coins be disentangled by . .
Step 5: The device measures the state of the two quantum coins and sends (or ) as the message to the arbitrator if the collapsed state of quantum coin is (or ).
Step 8: The arbitrator receives the overall message and assigns payoffs to the two agents according to Table 1. END.

In the A-EWL model, the assumed device performs quantum measurements and sends messages to the arbitrator on behalf of agents. Thus, the arbitrator needs not work with an additional quantum equipment as EWL model requires, i.e., the arbitrator works in the same way as before. It should be emphasized that the A-EWL model does not aim to solve the criticisms on the EWL model as specified in the Introduction. We propose the A-EWL model only for the following simulation process, which is a key part of the algorithmic model.

Since quantum operations can be simulated classically by using complex numbers, the A-EWL model can also be simulated. In what follows we will give matrix representations of quantum states and then propose an algorithmic version of A-EWL model.

### 2.2 Matrix representations of quantum states

In quantum mechanics, a quantum state can be described as a vector. For a two-level system, there are two basis vectors: and . In the beginning, we define:

 |CC⟩=[1,0,0,0]T,|CD⟩=[0,1,0,0]T,|DC⟩=[0,0,1,0]T,|DD⟩=[0,0,0,1]T.
 ^J=⎡⎢ ⎢ ⎢ ⎢⎣cos(γ/2)00isin(γ/2)0cos(γ/2)isin(γ/2)00isin(γ/2)cos(γ/2)0isin(γ/2)00cos(γ/2)⎤⎥ ⎥ ⎥ ⎥⎦,γ∈[0,π/2].

For ,

 ^Jπ/2=1√2⎡⎢ ⎢ ⎢⎣100i01i00i10i001⎤⎥ ⎥ ⎥⎦,^J+π/2=1√2⎡⎢ ⎢ ⎢⎣100−i01−i00−i10−i001⎤⎥ ⎥ ⎥⎦,

where is the conjugate of .

Definition 1: .
Since only two values in are non-zero, we only need to calculate the leftmost and rightmost column of to derive .

Definition 2: .

Suppose , let . It can be easily checked that , , and are all unitary matrices. Hence, . Thus, can be viewed as a probability distribution over the states .

### 2.3 An algorithmic model

Based on the matrix representations of quantum states, here we will propose an algorithmic model that simulates the A-EWL model. Since the entanglement measurement is a control factor, it can be simply set as its maximum . The input and output of the algorithmic model are shown in Fig. 2. A Matlab program is shown in Fig. 3(a)-(d).

Input:
1) , : the parameters of agent ’s local operation , .
2) , : the messages written on the two sides of agent ’s card. and represent Cooperate and Defect respectively.

Output:
{}, : agent ’s message that is sent to the arbitrator.

Procedures of the algorithmic model:
Step 1: Reading two parameters and from each agent (See Fig. 3(a)).
Step 2: Computing the leftmost and rightmost columns of (See Fig. 3(b)).
Step 3: Computing , , and the probability distribution (See Fig. 3(c)).
Step 4: Randomly choosing a state from the set of all four possible states according to the probability distribution .
Step 5: For each , the computer sends (or ) as message to the arbitrator through channel if the -th element of the chosen state is (or ) (See Fig. 3(d)).

## 3 Five types of one-shot PD

Since its beginning, PD has been generalized to many disciplines such as politics, economics, sociology, biology and so on. Despite these widespread applications, people seldom care how the payoffs of agents are determined. For example, Axelrod [1] used the word “yield” to describe how the agents obtained the payoffs. Nowak and May [2] used the word “get”, and Santos and Pacheco [3] used the word “receive” respectively.

One may think that such question looks trivial at first sight. However, as we will show in this section, there exists an interesting story behind this question. In what follows, we will categorize the one-shot PD into five different types.

Type-1 PD:
1) There are two agents and no arbitrator in the game.
2) The strategies of agents are actions performed by agents. The agents’ payoffs are determined by the outcomes of these actions and satisfy Table 1.

For example, let us neglect the United Nation and consider two countries (e.g., US and Russia) confronted the problem of nuclear disarmament. The strategy Cooperate means “Obeying disarmament”, and Defect means “Refusing disarmament”. If the payoff matrix confronted by the two countries satisfies Table 1, the nuclear disarmament game is a type-1 PD.

Type-2 PD:
1) There are two agents and an arbitrator in the game.
2) The strategies of agents are actions performed by agents. The arbitrator observes the outcomes of actions and assign payoffs to the agents according to Table 1.

For example, let us consider a taxi game. Suppose there are two taxi drivers and a manager. Two drivers drive a car in turn, one in day and the other in night. The car’s status will be very good, ok or common if the number of drivers who maintain the car is two, one or zero respectively. The manager observes the car’s status and assigns rewards , , to each driver respectively, where . The whole cost of maintenance is . Let the strategy Cooperate denote “Maintain”, and Defect denote “Not maintain”. The payoff matrix can be represented as Table 2. If Table 2 satisfies the conditions in Table 1, the taxi game is a type-2 PD.

Table 2: The payoff matrix of type-2 PD.
\backslashboxagent 1agent 2 Cooperate Defect Cooperate () () Defect () ()

Type-3 PD:
1) There are two agents and an arbitrator in the game.
2) The strategy of each agent is not an action, but a message that can be sent to the arbitrator through a channel. The arbitrator receives two messages and assign payoffs to the agents according to Table 1.
3) Two agents cannot communicate with each other.

For example, suppose two agents are arrested separately and required to report their crime information to the arbitrator through two channels independently. If the arbitrator assigns payoffs to agents according to Table 1, this game is a type-3 PD.

Type-4 PD:
Conditions 1-2 are the same as those in type-3 PD.
3) Two agents can communicate with each other.
4) Before sending messages to the arbitrator, two agents can construct the algorithmic model specified in Fig. 2. Each agent can observe whether the other agent participates the algorithmic model or not: whenever the other agent takes back his channel, agent will do so and sends his message to the arbitrator directly.

Remark 1: At first sight, the conditions of type-4 PD is complicated. However, these conditions are not restrictive when the arbitrator communicate with agents indirectly and cannot separate them. For example, suppose the arbitrator and agents are connected by Internet, then all conditions of type-4 PD can be satisfied in principle.

The type-4 PD works in the following way:
Stage 1: (Actions of two agents) For each agent , he faces two strategies:
: Participate the algorithmic model, i.e., leave his channel to the computer, and submit to the computer;
: Not participate the algorithmic model, i.e., take back his channel, and submit to the arbitrator directly.
According to condition 4, the algorithmic model is triggered if and only if both two agents participate it.
Stage 2: (Actions of the arbitrator) The arbitrator receives two messages and assigns payoffs to agents according to Table 1.

In type-4 PD, from the viewpoints of the arbitrator, he acts in the same way as before, i.e., nothing is changed. However, the payoff matrix confronted by two agents is now changed to Table 3. For each entry of Table 3, we give the corresponding explanation as follows:

Table 3: The payoff matrix of two agents by constructing the algorithmic model, where are defined in Table 1, .
\backslashboxagent 1agent 2 (R, R) (P, P) (P, P) (P, P)

1) : This strategy profile means two agents both participate the algorithmic model and submit parameters to the computer. According to Ref. [5], for each agent , his dominant parameters are and , which result in a Pareto-efficient payoff .
2) : 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 is , and the arbitrator will assign the Pareto-inefficient payoff to agents.
3) : This strategy profile is similar to the above case. The arbitrator will assign to two agents.
4) : 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 to two agents.

From Table 3, it can be seen that and are two Nash equilibria, and the former is Pareto-efficient. As specified by Telser (Page 28, Line 2, [11]), “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 is a self-enforcing Nash equilibrium and the Pareto-efficient payoff 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 cannot escape the dilemma.

## 4 Discussions

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 cannot 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 , 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 , and generates the Pareto-efficient payoff 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 , , , in the algorithmic model respectively, since , .
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 cannot 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 cannot 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.

## 5 Conclusion

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.

## Acknowledgments

The author is very grateful to Ms. Fang Chen, Hanyue Wu (Apple), Hanxing Wu (Lily) and Hanchen Wu (Cindy) for their great support.

## References

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• [5] J. Eisert, M. Wilkens, M. Lewenstein, Quantum games and quantum strategies, Phys. Rev. Lett., 83 (1999) 3077-3080.
• [6] J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou and R. Han, Experimental realization of quantum games on a quantum computer, Phys. Rev. Lett., 88 (2002) 137902.
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