Subjective expected relative similarity
Subjective expected relative similarity
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 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 at the University of Haifa.
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 <math>T > R > P > S</math> (and, <math display="inline">R \geq \frac{T+S}{2}</math> 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 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 of the game. However, this is a deficient equilibrium (since mutual cooperation results in a better payoff for both players).
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.
The SERS theory SERS assumes that the similarity between the players is subjectively and individually perceived (denoted as <math>p_s</math>, where <math>0\leq p_s\leq 1</math>). 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 <math>p_s</math>, 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 of cooperation is given by <math>R\cdot p_s + S\cdot (1 - p_s) </math> and the expected payoff of defection is given by <math>P\cdot p_s + T\cdot (1 - p_s)</math>. Hence, cooperation provides a higher expected payoff whenever <math>R\cdot p_s + S\cdot (1 - p_s) > P\cdot p_s + T\cdot (1 - p_s)</math> which may also be expressed further as:
Cooperate if <math display="inline">p_s>\frac{T-S}{T-S+R-P}</math>. Defining <math display="inline">p_s^=\frac{T-S}{T-S+R-P}</math>, we obtain a simple decision rule: cooperate whenever <math>p_s > p_s^</math>, where <math>p_s</math> denotes the level of perceived similarity with the opponent, and <math>p_s^*</math> denotes the similarity threshold derived from the payoff matrix.
To illustrate, consider a PD payoff matrix with <math>T = 5, R = 3, P = 1, S = 0</math>. The similarity threshold calculated for the game is given by: <math>p_s^*= \frac{5 - 0}{5 - 0 + 3 - 1}\approx 0.71</math>. Thus a player perceiving the similarity with the opponent, <math>p_s</math>, exceeding 0.71 should cooperate in order to maximise his expected payoffs.