1. When fairness bends rationality: Ernst Fehr meets John Nash
Alessandro Tavoni
Preamble: from efficiency to equality
Incorporating fairness in stationary models
Fairness in evolutionary models
By allowing for IA preferences, cooperation can be established: sufficient conditions for cooperation to be Evolutionarily Stable (ES), risk-dominant (RD) and advantageous (AD) are characterized in terms of the payoffs (b,c)
The Nash Equilibrium (NE) is a game theory concept that assumes that individuals rationally maximize their payoff, disregarding distributive concerns over the payoffs of other players
Two concepts are proposed to move away from the Nash equilibrium shortcomings: IBE+IA and QRE+IA
Model
Game1
Game2
Game3
Game4
Game 5 6
Mean NE
6.076
1.225
.354
.708
.422
.064
1.475
IBE
.315
.035
.416
.224
.094
.205
.215
IBE+IA
.746
.178
.428
.152
.140
.030
.279
QRE+IA
.251
.012
.397
.036
.163
.027
.147
Substantial evidence has accumulated in recent empirical works on the limited ability of the Nash equilibrium to rationalize observed behavior in many classes of games are we missing inequity aversion?
COOPERATION is: C D
b-c
-c-α(b+c)
b-β(b+c)
ES:
Game theory revisited
The idea behind this work is to utilize established game theory concepts such as Impulse Balance and Quantal Response, but introducing inequity aversion parameters: how much am I making relative to you?
RD:
AD:
Figure 1. Mean Square Deviations of the proposed concepts
Figure 3. IA explains the survival of cooperation in PD game
Impulse Balance Equilibrium (IBE) is a stationary concept based on the idea that agents compare the payoff achieved in a game to an aspiration level, and experience an impulse towards better-performing strategies
Quantal Response Equilibrium (QRE) is a probabilistic choice model where individuals are more likely to select better choices than worse choices, but do not necessarily succeed in selecting the very best choice
Inequality Aversion (IA) is a generalization of the traditional utility function to model resistance to inequitable outcomes. People may be willing to give up some payoff in order to move in the direction of more equitable outcomes:
A disutility (Envy) arises when Altruistic disutility
the material payoffs of the other arises when the
player exceeds my own: α> other player’s payoff
is below my own: β>0
Evolutionary escape from the Prisoner’s Dilemma (PD)
PD sets the harshest conditions for emergence and survival of cooperation. Let b=benefit and c=cost associated with the cooperative act
Conclusions
Inequity aversion plays a crucial role in affecting individual behavior. In particular:
Artificial data on behavior generated by the IA hybrid models fits closely the experimental data
Whether a cooperator can invade or not a population of defectors boils down to a comparison of the sign and magnitude of α and β (the parameters modelling agents’ inequity aversion) C D
b-c -c b
Acknowledgments
Levin Lab, EEB department, Princeton University
Massimo Warglien, Advanced school of Economics
References
FEHR, E., SCHMIDT, K. (1999): A theory of fairness, competition and cooperation, Quarterly journal of Economics
MCKELVEY, RICHARD, D., PALFREY, THOMAS, R. (1995): Quantal Response Equilibria for Normal Form Games, Games and Economic Behavior; 10:1
SELTEN, R., CHMURA, T. (2008): Stationary Concepts for Experimental 2x2 Games, American Economic Review, 98:3
Figure 2. PD game and associated payoffs to i cooperators
Without a mechanism for the evolution of cooperation, selection will drive the number of cooperators i (and the average fitness of the population) to 0 . But we know people cooperate are we missing the IA part?