The **Optional Prisoner's Dilemma** (OPD) game models a situation of conflict involving two players in game theory. It can be seen as an extension of the standard prisoner's dilemma game, where players have the option to "reject the deal", that is, to abstain from playing the game.^{ [1] } This type of game can be used as a model for a number of real world situations in which agents are afforded the third option of abstaining from a game interaction such as an election. ^{ [2] }

The structure of the Optional Prisoner's Dilemma can be generalized from the standard prisoner's dilemma game setting. In this way, suppose that the two players are represented by the colors, red and blue, and that each player chooses to "Cooperate", "Defect" or "Abstain". ^{ [3] }

The payoff matrix for the game is shown below:

Cooperate | Defect | Abstain | |
---|---|---|---|

Cooperate | R, R | S, T | L, L |

Defect | T, S | P, P | L, L |

Abstain | L, L | L, L | L, L |

- If both players cooperate, they both receive the reward
*R*for mutual cooperation. - If both players defect, they both receive the punishment payoff
*P*. - If Blue defects while Red cooperates, then Blue receives the temptation payoff
*T*, while Red receives the "sucker's" payoff,*S*. - Similarly, if Blue cooperates while Red defects, then Blue receives the sucker's payoff
*S*, while Red receives the temptation payoff*T*. - If one or both players abstain, both receive the loner's payoff
*L*.

The following condition must hold for the payoffs:

*T* > *R* > *L* > *P* > *S*

An **evolutionarily stable strategy** (**ESS**) is a strategy which, if adopted by a population in a given environment, is impenetrable, meaning that it cannot be invaded by any alternative strategy that are initially rare. It is relevant in game theory, behavioural ecology, and evolutionary psychology. An ESS is an equilibrium refinement of the Nash equilibrium. It is a Nash equilibrium that is "evolutionarily" stable: once it is fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from invading successfully. The theory is not intended to deal with the possibility of gross external changes to the environment that bring new selective forces to bear.

* The Evolution of Cooperation* is a 1984 book by political scientist Robert Axelrod that expanded a highly influential paper of the same name, and popularized the study upon which the original paper had been based.

The **prisoner's dilemma** is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it "prisoner's dilemma", presenting it as follows:

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:

In game theory, the **Nash equilibrium**, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

**Tit for tat** is an English saying meaning "equivalent retaliation". It developed from "tip for tap", first used in 1558.

In economics and game theory, a participant is considered to have **superrationality** if they have perfect rationality but assume that all other players are superrational too and that a superrational individual will always come up with the same strategy as any other superrational thinker when facing the same problem. Applying this definition, a superrational player playing against a superrational opponent in a prisoner's dilemma will cooperate while a rationally self-interested player would defect.

The **game of chicken**, also known as the **hawk–dove game** or **snowdrift game**, is a model of conflict for two players in game theory. The principle of the game is that while the outcome is ideal for one player to yield, but the individuals try to avoid it out of pride for not wanting to look like a 'chicken'. So each player taunts the other to increase the risk of shame in yielding. However, when one player yields, the conflict is avoided, and the game is for the most part over.

In game theory, the **best response** is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.

**Evolutionary game theory** (**EGT**) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.

In game theory, the **centipede game**, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.

In game theory, **grim trigger** is a trigger strategy for a repeated game.

In game theory, the **stag hunt** is a game that describes a conflict between safety and social cooperation. Other names for it or its variants include "assurance game", "coordination game", and "trust dilemma". Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, they must have the cooperation of their partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag. This has been taken to be a useful analogy for social cooperation, such as international agreements on climate change.

In game theory, a **solution concept** is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

In game theory, **normal form** is a description of a *game*. Unlike extensive form, normal-form representations are not graphical *per se*, but rather represent the game by way of a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, for each player.

The **public goods game** is a standard of experimental economics. In the basic game, subjects secretly choose how many of their private tokens to put into a public pot. The tokens in this pot are multiplied by a factor and this "public good" payoff is evenly divided among players. Each subject also keeps the tokens they do not contribute.

The **chainstore paradox** is an apparent game theory paradox involving the chain store game, where a "deterrence strategy" appears optimal instead of the backward induction strategy of standard game theory reasoning.

**Peace war game** is an iterated game originally played in academic groups and by computer simulation for years to study possible strategies of cooperation and aggression. As peace makers became richer over time it became clear that making war had greater costs than initially anticipated. The only strategy that acquired wealth more rapidly was a "Genghis Khan", a constant aggressor making war continually to gain resources. This led to the development of the "provokable nice guy" strategy, a peace-maker until attacked. Multiple players continue to gain wealth cooperating with each other while bleeding the constant aggressor.

In game theory, the **traveler's dilemma** is a non-zero-sum game in which each player proposes a payoff. The lower of the two proposals wins; the lowball player receives the lowball payoff plus a small bonus, and the highball player receives the same lowball payoff, minus a small penalty. Surprisingly, the Nash equilibrium is for both players to aggressively lowball. The traveler's dilemma is notable in that naive play appears to outperform the Nash equilibrium; this apparent paradox also appears in the centipede game and the finitely-iterated prisoner's dilemma.

The **volunteer's dilemma** game models a situation in which each player can either make a small sacrifice that benefits everybody, or instead wait in hope of benefiting from someone else's sacrifice.

**Subjective expected relative similarity (SERS)** is a normative and descriptive theory that predicts and explains cooperation levels in a family of games termed *Similarity Sensitive Games (SSG)*, among them the well-known Prisoner's Dilemma game (PD). SERS was originally developed in order to (i) provide a new rational solution to the PD game and (ii) to predict human behavior in single-step PD games. It was further developed to account for: (i) repeated PD games, (ii) evolutionary perspectives and, as mentioned above, (iii) the SSG subgroup of 2x2 games. SERS predicts that individuals cooperate whenever their subjectively perceived similarity with their opponent exceeds a situational index derived from the game’s payoffs, termed the similarity threshold of the game. SERS proposes a solution to the rational paradox associated with the single step PD and provides accurate behavioral predictions. The theory was developed by Prof. Ilan Fischer at the University of Haifa.

- ↑ Cardinot, Marcos; Gibbons, Maud; O'Riordan, Colm; Griffith, Josephine (2016). "Simulation of an Optional Strategy in the Prisoner's Dilemma in Spatial and Non-spatial Environments".
*From Animals to Animats 14*. Lecture Notes in Computer Science.**9825**. pp. 145–156. doi:10.1007/978-3-319-43488-9_14. ISBN 978-3-319-43487-2. - ↑ Batali, John; Kitcher, Philip (1995). "Evolution of altriusm in optional and compulsory games" (PDF).
*Journal of Theoretical Biology*.**175**(2): 161–171. doi:10.1006/jtbi.1995.0128. - ↑ Cardinot, Marcos; O'Riordan, Colm; Griffith, Josephine (2016). "The Optional Prisoner's Dilemma in a Spatial Environment: Coevolving Game Strategy and Link Weights".
*ECTA*. Proceedings of the 8th International Joint Conference on Computational Intelligence.**1**. pp. 86–93. doi:10.5220/0006053900860093. ISBN 978-989-758-201-1.

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