A Method to Select Best Among Multi-Nash Equilibria

A non-zero sum bimatrix game may yield numerous Nash equilibrium solutions while solving the game. The selection of a good Nash equilibrium from among the many options poses a dilemma. In this article, three methods have been proposed to select a good Nash equilibrium. The first approach identifies the most payoff-dominant Nash equilibrium, while the second method selects the most risk-dominant Nash equilibrium. The third method combines risk dominance and payoff dominance by giving due weights to the two criteria. A sensitivity analysis is performed by changing the relative weights of criteria to check its effect on the ranks of the multiple Nash equilibria, infusing more confidence in deciding the best Nash equilibrium. JEL Codes: C7, C72, D81

Nash versus coarse correlation

We run a laboratory experiment to test the concept of coarse correlated equilibrium (Moulin and Vial in Int J Game Theory 7:201–221, 1978), with a two-person game with unique pure Nash equilibrium which is also the solution of iterative elimination of strictly dominated strategies. The subjects are asked to commit to a device that randomly picks one of three symmetric outcomes (including the Nash point) with higher ex-ante expected payoff than the Nash equilibrium payoff. We find that the subjects do not accept this lottery (which is a coarse correlated equilibrium); instead, they choose to play the game and coordinate on the Nash equilibrium. However, given an individual choice between a lottery with equal probabilities of the same outcomes and the sure payoff as in the Nash point, the lottery is chosen by the subjects. This result is robust against a few variations. We explain our result as selecting risk-dominance over payoff dominance in equilibrium.

Impulse Response Dynamics in Weakest Link Games

In a recent paper, Croson et al. (2015) experimentally study three weakest link games with multiple symmetric equilibria. They demonstrate that static concepts based on the Nash equilibrium (including multiple Nash equilibria, quantal response equilibria, and equilibrium selection by risk and payoff dominance) cannot successfully capture the observed treatment differences. Using Reinhard Selten’s impulse response dynamics, we derive a proposition that provides a theoretical ranking of contribution levels in the weakest link games. We show that the predicted ranking of treatment outcomes is fully consistent with the observed data. In addition, we demonstrate that the impulse response dynamics perform well in tracking average contributions over time. We conclude that Reinhard Selten’s impulse response dynamics provide an extremely valuable behavioral approach that is not only capable of resolving the indecisiveness of static approaches in games with many equilibria, but that can also be used to track the development of choices over time in games with repeated interaction.

Modified UCT Algorithm with Risk Dominance Methods in Imperfect Information Game

UCT (Upper confidential bounds on Trees) has been applied quite well as a selection approach in MCTS(Monte Carlo Tree Search) in imperfect information games like poker. By using risk dominance as complementary part of decision method besides payoff dominance, opponent strategies is better characterized as their risk factors, like bluff actions in Texas Hold’em Poker . In this paper, estimation method about the influence of risk factors on computing game equilibrium is provided. A novel algorithm, UCT-risk is proposed as modification about UCT algorithm basing on risk estimation methods. To verify the performance of new algorithm, Texas Hold’em, a popular test-bed for AI research is chosen as the experiment platform. The Agent adopted UCT-risk algorithm performs as well or better as the best previous approaches in experiments. And also it is applied in a poker agent named HITSZ_CS_13 in the 2013 AAAI Computer Poker Competition, which confirms the effectiveness of the UCT-risk provided in this paper.

Love is not enough: Other-regarding preferences cannot explain payoff dominance in game theory

Even if game theory is broadened to encompass other-regarding preferences, it cannot adequately model all aspects of interactive decision making. Payoff dominance is an example of a phenomenon that can be adequately modeled only by departing radically from standard assumptions of decision theory and game theory – either the unit of agency or the nature of rationality.