Written by: Tom Garcia Professor (retired)
Booth School of Business 01/29/19
In John Nash’s classical formulation of a non-cooperative game involving two or more players , each player is assumed to know the equilibrium strategies of the other players. Among the many studies since, in a paper co-authored with Bill Zangwill , we reinvestigate an obvious relaxation of Nash’s assumption, first proposed in [3, 4], that more accurately reflects real world situations: what if the players’ strategies are not common knowledge, but rather that a player only has subjective beliefs of the other players’ strategies?
Using Bayesian analysis, we discovered the unique solution to this reformulated game. Our solution, when applied to the more than a thousand year old rock-paper-scissors game, is new, as far as we know, but obvious once stated: play rock (paper, scissors) if you believe that your opponent will play paper (scissors, rock) with probability of at most one-third and will play scissors (rock, paper) with probability at least one-third.
The above solution divides the 3D Cartesian plane (or the 2D unit simplex) into 6 regions, where the play is prescribed in each region. (Please refer to the table below. Two regions are crossed out because the sum of the probabilities must equal one.) If players’ beliefs are common knowledge, then the above solution shortens to the Nash solution (1/3, 1/3, 1/3). Otherwise, if say, your belief regarding your opponent prescribes that you play rock, then your opponent, knowing your belief, will play paper, which is incompatible to your belief.
Suppose you have a history of your opponent’s plays of the game. Using known statistical methods, you can judge if your opponent plays randomly. (Most humans do not play randomly, and if they do, their attempts to generate random numbers are not mathematically random.) If your opponent appears not to be a random player, you may be at an advantage if you use AI methods for judging which of the table’s 6 regions your opponent will likely to be in.
- Nash, J (1950) Equilibrium points in n-person games. Proceedings of the National Academy of Sciences 36(1):48-49
- Garcia CB, Zangwill WI (2017) A New Foundation for Game Theory. Working paper
- Harsanyi J (1967) Games With Incomplete Information Played by “Bayesian” Players I – III. J. Management Science 14 (3):159-182
- Kadane JB, Larkey PD (1982) Subjective Probability and the Theory of Games. Management Science 28 (2):113-120