An evolutionary approach to social choice problems with q-quota rules (with Akira Okada) [pdf]
This paper considers a dynamic process of n-person social choice problems under q-majority where a status-quo policy is challenged by an opposing policy drawn randomly in each period. The opposing policy becomes the next status-quo if it receives at least q votes. We characterize stochastically stable policies under a boundedly rational choice rule of voters. Under the best response rule with mutations, a Condorcet winner is stochastically stable for all q-quota rules, and uniquely so if q is greater than the min-max quota. Under the logit choice rule, the Borda winner is stochastically stable under the unanimity rule. Our evolutionary approach provides a dynamic foundation of the mini-max policies in multidimensional choice problems with Euclidean preferences. .
Prospect theory Nash bargaining solution and its stochastic stability [pdf]
We consider stochastic stability with players obeying prospect theory. We extend Young’s evolutionary bargaining model to a two-stage Nash demand game where two players simultaneously choose whether to exercise an outside option in the first stage, and they play the Nash demand game in the second stage, which will be reached only if no player exercises their option. In this setting, the value of the option naturally serves as a reference point for the players. We address the dependence of stochastically stable divisions on reference points, and show that those divisions constantly differ from the Nash bargaining solution under expected utility theory. Inspired by this, we propose prospect theory Nash bargaining solution, which coincides with the stochastically stable division.
Stochastic Stability in the Large Population and Small Mutation Limits for General Normal Form Games [pdf]
We consider models of stochastic evolution in normal form games with K>=2 strategies in which agents employ the best response with mutations choice rule. Forming the dynamics as a Markov chain with state space being the set of best responses, we overcome technical difficulties which arises with the large population. We study the long run behavior of the Markov chain for four limiting cases: the small noise limit, the large population limit, and the both orders of the double limit. We characterize conditions under which selection results of the both orders of the double limit coincide..
This paper examines a dynamic process of n-person coalitional bargaining problems. We study the stochastic evolution of social conventions by embedding a static bargaining setting in a dynamic process; Over time agents revise their coalitions and surplus distributions under the presence of stochastic noise which leads agents to make a suboptimal decision. Under a logit specification of choice probabilities, we find that the stability of a core allocation decreases in the wealth of the richest player, and that stochastically stable allocations are core allocations which minimize the wealth of the richest.
This paper investigates the role of loss-aversion in affecting the long run equilibria of stochastic evolutionary dynamics. We consider a finite population of loss-averse agents who are repeatedly and randomly matched to play a 2 x 2 coordination game. When an agent revises her strategy, she compares the payoff from each strategy to a reference point which is endogenously formed. Based on the comparison, she makes a (possibly stochastic) choice. We call the resulting dynamics prospect dynamics. Three types of endogenous reference points are examined: social average, expectations and status-quos. We find that risk-dominance is no longer sufficient to guarantee stochastic stability under prospect dynamics with any type of reference points. Therefore, we propose a stronger concept, loss-dominance: a strategy is loss-dominant if it is both risk-dominant strategy and the maximin strategy. This concept captures people's psychological needs to avoid not only risks but also losses. We show that it serves as a natural selection refinement for games with loss-averse agents. The state in which all agents play the loss-dominant strategy (if exists) is uniquely stochastically stable under prospect dynamics for any degree of loss-aversion and all types of reference points. We also characterize the precise conditions for stochastic stability in games with no loss-dominant strategy.
This paper considers marriage problems, roommate problems with nonempty core, and college admissions problems with responsive preferences. All stochastically stable matchings are shown to be contained in the set of matchings which are most robust to one-shot deviation.
Journal of Economic Theory 154, pp.562--577 (2014)
We consider deterministic evolutionary dynamics under imitative revision protocols. We allow agents to have different aspiration levels in the imitative protocols where their aspiration levels are not observable to other agents. We show that the distribution of strategies becomes statistically independent of the aspiration level eventually in the long run. Thus, long-run properties of homogeneous imitative dynamics hold as well, despite heterogeneity in aspiration levels.
Games and Economic Behavior 88, pp.90--111 (2014)
This paper examines a dynamic process of unilateral and joint deviations of agents and the resulting stochastic evolution of social conventions. Our model unifies stochastic stability analysis in static settings, including normal form games, network formation games, and simple exchange economies, as stochastic stability analysis in a class of interactions in which agents unilaterally and jointly choose their strategies. We embed a static setting in a dynamic process; Over time agents revise their strategies based on the improvements that the new strategy profile offers them. In addition to the optimization process, there are persistent random shocks on agents utility that potentially lead to switching to suboptimal strategies. Under a logit specification of choice probabilities, we characterize the set of states that will be observed in the long-run as noise vanishes. We apply these results to examples of certain potential games.
International Journal of Game Theory 41, pp.489--496 (2012)
Bergin and Lipman (1996) show that equilibrium selection using stochastic evolutionary processes depends on the specification of mutation rates. We offer a characterization of how mutation rates determine the selection of Nash equilibria in 2x2 symmetric coordination games for single and double limits of the small mutation rates and the large population size.
Copyright 2012 Ryoji Sawa
University of Aizu