Subjective Expected Relative Similarity
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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 The prisoner's dilemma is a game theory thought experiment involving two rational agents, each of whom can either cooperate for mutual benefit or betray their partner ("defect") for individual gain. The dilemma arises from the fact that while def ...
game (PD).Fischer, I. (2012). Similarity or reciprocity? On the determinants of cooperation in similarity-sensitive games. Psychological Inquiry, 23(1), 48-54. 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 2×2 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 Ilan may refer to: Organization *ILAN, Israeli umbrella organization for the treatment of disabled children Given name *Ilan (name), a Hebrew/Israeli name *Ilan Bakhar, a retired Israeli footballer *Ilan Araújo Dall'Igna, a Brazilian footballer ...
at the
University of Haifa The University of Haifa (, ) is a public research university located on Mount Carmel in Haifa, Israel. Founded in 1963 as a branch of the Hebrew University of Jerusalem, the University of Haifa received full academic accreditation as an inde ...
.


The Prisoner's Dilemma

The dilemma is described by a 2 × 2 payoff matrix that allows each player to choose between a cooperative and a competitive (or defective) move. If both players cooperate, each player obtains the reward (R) payoff. If both defect, each player obtains the punishment (P) payoff. However, if one player defects while the other cooperates, the defector obtains the temptation (T) payoff and the cooperator obtains the sucker's (S) payoff, where T > R > P > S (and, R \geq \frac assuring that sharing the payoffs awarded for uncoordinated choices does not exceed the payoffs obtained by mutual cooperation). Given the payoff structure of the game (see Table 1), each individual player has a
dominant strategy In game theory, a strategy ''A'' dominates another strategy ''B'' if ''A'' will always produce a better result than ''B'', regardless of how any other player plays. Some very simple games (called straightforward games) can be solved using domi ...
of defection. This dominant strategy yields a better payoff regardless of the opponent's choice. By choosing to defect, players protect themselves from exploitation and retain the option to exploit a trusting opponent. Because this is the case for both players, mutual defection is the only
Nash equilibrium In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...
of the game. However, this is a deficient equilibrium (since mutual cooperation results in a better payoff for both players).Fischer, I. (2009). Friend or foe: subjective expected relative similarity as a determinant of cooperation. Journal of Experimental Psychology: General, 138(3), 341. The PD game payoff matrix:


The repeated prisoner's dilemma

Players that knowingly interact for several games (where the end point of the game is unknown), thus playing a repeated Prisoner's Dilemma game, may still be motivated to cooperate with their opponent while attempting to maximise their payoffs along the entire set of their repeated games. Such players face a different challenge of choosing an efficient and lucrative strategy for the repeated play. This challenge may become more complex when individuals are embedded in an ecology, having to face many opponents with various and unknown strategies.Nowak, M., & Sigmund, K. (1993). A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game. Nature, 364(6432), 56-58.


The SERS theory

SERS assumes that the similarity between the players is subjectively and individually perceived (denoted as p_s, where 0\leq p_s\leq 1). Two players confronting each other may have either identical or different perceptions of their similarity to their opponent. In other words, similarity perceptions need neither be symmetric nor correspond to formal logic constraints. After perceiving p_s, each player chooses between cooperation and defection, attempting to maximize the expected outcome. This means that each player estimates his or her expected payoffs under each of two possible courses of action. The
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of cooperation is given by R\cdot p_s + S\cdot (1 - p_s) and the expected payoff of defection is given by P\cdot p_s + T\cdot (1 - p_s). Hence, cooperation provides a higher expected payoff whenever R\cdot p_s + S\cdot (1 - p_s) > P\cdot p_s + T\cdot (1 - p_s) which may also be expressed further as: Cooperate if p_s>\frac. Defining p_s^*=\frac, we obtain a simple
decision rule In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy in game ...
: cooperate whenever p_s > p_s^*, where p_s denotes the level of perceived similarity with the opponent, and p_s^* denotes the similarity threshold derived from the payoff matrix. To illustrate, consider a PD payoff matrix with T = 5, R = 3, P = 1, S = 0. The similarity threshold calculated for the game is given by: p_s^*= \frac\approx 0.71. Thus a player perceiving the similarity with the opponent, p_s, exceeding 0.71 should cooperate in order to maximise his expected payoffs.


Empirical evidence

Several experiments were conducted to test whether SERS provides not only a normative theory but also a descriptive theory of human behaviour.
For example, an experiment involving 215 university undergraduates revealed an average of 30% cooperation rate for a payoff matrix with p_s^* = 0.8and an average of 46% cooperation rate for a payoff matrix p_s^* = 0.63. Participants cooperated 47% under high level of induced similarity and only 29% under low level of induced similarity. The cooperation rate for manipulating the perception of similarity of the opponent, revealed an increase from 67% to 80% of cooperation for the lower similarity threshold and from 40% to 70% cooperation for the higher similarity threshold.
Other experiments with various similarity induction methods and payoff matrices further confirmed SERS's status as a descriptive theory of human behaviour.


The SERS theory for Repeated PD Games

Experiments on the impact of SERS on repeated games are presently being conducted and analysed at the University of Haifa and the
Max Planck Institute for Research on Collective Goods The Max Planck Institute for Research on Collective Goods ( German: ''Max-Planck-Institut zur Erforschung von Gemeinschaftsgütern'') is located in Bonn, Germany Germany, officially the Federal Republic of Germany, is a country in Central ...
in Bonn.


Similarity sensitive games

The PD game is not the only similarity sensitive game. Games for which the choice of the action with the higher expected value depends on the value of p_s are defined as Similarity Sensitive Games (SSGs), whereas others are nonsimilarity sensitive. Focusing only on the 24 completely rank-ordered and symmetric games, we can mark 12 SSGs. After eliminating games that reflect permutations of other games generated either by switching rows, columns, or both rows and columns, we are left with six basic (completely rank-ordered and symmetric) SSGs. These are games for which SERS provides a rational and payoff-maximizing strategy that recommends which alternative to choose for any given perception of similarity with the opponent.


Mimicry and Relative Similarity (MaRS)

Developing the SERS theory into an evolutionary strategy yields the ''Mimicry and Relative Similarity (MaRS)'' algorithm. Fusing enacted and expected mimicry generates a powerful and cooperative mechanism that enhances fitness and reduces the risks associated with trust and cooperation. When conflicts take the form of repeated PD games, individuals get the opportunity to learn and monitor the extent of similarity with their opponents. They can then react by choosing whether to enact, expect, or exclude mimicry. This rather simple behavior has the capacity to protect individuals from exploitation and drive the evolution of cooperation within entire populations. MaRS paves the way for the induction of cooperation and supports the survival of other cooperative strategies. The existence of MaRS in heterogeneous populations helps those cooperative strategies that do not have the capacity of MaRS to combat hostile and random opponents. Despite the fact that MaRS cannot prevail in a duel with an unconditional defector, interacting within heterogeneous populations allows MaRS to fight unpredictable and hostile strategies and cooperate with cooperative ones, including itself. The operation of MaRS promotes cooperation, minimizes the extent of exploitation, and accounts for high fitness levels. Testing the model in computer simulations of behavioral niches, populated with agents that enact various strategies and learning algorithms, shows how mimicry and relative similarity outperforms all the opponent strategies it was tested against, pushes noncooperative opponents toward extinction, and promotes the development of cooperative populations.Fischer, I., Frid, A., Goerg, S. J., Levin, S. A., Rubenstein, D. I., & Selten, R. (2013). Fusing enacted and expected mimicry generates a winning strategy that promotes the evolution of cooperation. Proceedings of the National Academy of Sciences, 110(25), 10229-10233.


See also

*
Game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...
*
Nash Equilibrium In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...
*
Tit for Tat Tit for tat is an English saying meaning "equivalent retaliation". It is an alternation of '' tip for tap'' "blow for blow", first recorded in 1558. It is also a highly effective strategy in game theory. An agent using this strategy will fi ...
* Win stay lose shift


References

{{Reflist Game theory