Bayes Rule in Card Games
When talking about updating strategies based on new information, card games like poker immediately come to mind. However, poker itself is a very complicated game. To study the effects of Bayes Rule, we can first consider a simple game, where players win if they can correctly discern whether their card is the highest out of all the players’ cards. The players can choose “Yes” if they think they have the highest card, or “No” if they do not. Without updates, where a player must reveal information sequentially, the probabilities are determined by the equation:
P [Z is Highest] = P [Higher than player X] x P [Higher than player Y | Higher than player X] (for a three person card game played by X, Y, and Z).
Let’s say we start with only one deck, and the cards are distributed to X, Y, and Z as follows; 9 is assigned to Z, 12 is assigned to X, and 10 is assigned to Y. Now, since Z doesn’t know what cards X and Y have, the probability he perceives of being higher than X is 8/12, since only 8 cards of the unknown 12 are less than 9. The probability that Z is higher than Y given that Z is higher than X is 7/11, because now Z only has 11 unknowns, 7 of which are less than 9. The probability that Z is the highest, then, is 8/12 x 7/11 = 0.42. Given these circumstances, Z would most likely choose “No”.
However, this is only simple probability simulation. To make a more interesting game, like in poker, we can also introduce the mechanic of announcing your bet or choice. In this case, the information that each player receives is the “Yes”/”No” choice of the player(s) before him.
As the article shows, previous choices give new information that increase or decrease the probability that players afterwards will win. In the example they discuss, John Nash uses the fact that Vanessa chooses “No” to determine that her card must be less than or equal to 9. He doesn’t know which of 8 possible cards it could be, but 7 of them are less than his value of 8. As we can see, his probability went from 7/12 to 7/8, just by knowing that Vanessa chose “No”.
I found it interesting that conditional probabilities were used to develop dominant strategies, before and after the update rule. Before the update, based on the probabilities of winning, a dominant strategy would be to choose “Yes” if your card is greater than or equal to 10, and “No” otherwise. However, after the update rule, the strategy was divided into three regions. At low values (1-8), the strategy of John Nash choosing “No” holds true regardless of what Vanessa does, which is similar to what happened without the update. At high values (12-13), also, John chooses “Yes” as he would without the update. The only real difference between the two games is in the middle range (9-11), where John’s best response depends on Vanessa’s new information. If she announces “No”, John is certain that her value is below his (or else her dominant strategy would have been to say “Yes”), so he will choose “Yes”. However, if she says “Yes”, John will know she has at least a 10, so he will say “No”.
Of course, this appears to put the first player at a disadvantage, since the other players get to know her choice, while she has nothing to go on. For that reason, the article introduces the concept of Bayesian-Nash Equilibrium, combining the concept of updated probability with Nash equilibrium strategies. With this, the first player can create his/her own best-response to maximize payoff by knowing how the other players will react. While these would be very hard to compute (especially payoffs and probabilities of certain strategies), the concept is useful to get a general idea of how high-performing poker players maximize payoff. Using probability to develop dominant strategies and best-responses is crucial in winning games like these, but at the same time, I feel like this concept could be applied to many other prediction techniques such as weather forecasts, stock options, and benefit-risk analysis.
(http://www.science4all.org/le-nguyen-hoang/bayesian-games-how-to-model-poker/)