It’s More Likely Your Opponent is Bluffing When It Comes to Large Bets in Poker
Poker is one of the many applications in which a mixed-strategy Nash Equilibrium arises. The article, “The Mathematics of Bluffing” provides an intuitive approach to understanding these equilibrium situations, and what it means for player’s tendencies to either call (i.e. match) an opponent’s bet or bluff in a river card situation of poker. To provide some background, the river card is considered the fifth and last card flipped by a dealer. Players are then allowed to offer their final wagers and, if no one folds, they must reveal their hands in a final showdown.
The poker game analyzed in this article is naturally simplified, but particularly applicable to the river card situation. Consider two players who are competing for a $1 pot. The probability that either player has the superior hand is random and modeled as 50% (or by the flip of a fair coin). Player 1 is allowed to look at the coin, and if he has heads, he knows he has a superior hand and vice versa. He can then either check (i.e. defer to player 2) or place a bet, and player 2 can respond by calling the bet or folding his hand. The payoff matrix for this situation is as follows:
| Strategies | P2 (Passive): Fold to any bet and call if P1 Checks | P2 (Aggressive): Call any bet by player 1 |
| P1 (Passive): Bet on heads, but check on tails | 0.5, 0.5 | 1, 0 |
| P1 (Aggressive): Bet on heads, but bluff on tails | 1, 0 | 0.5, 0.5 |
Source: http://www.huffingtonpost.com/druce-vertes-cfa/the-mathematics-of-bluffi_b_7989900.html
This example provides excellent intuition for why a mixed-strategy Nash Equilibrium arises, and why both players become indifferent to the strategy they use. For example, if player 1 decides to play aggressively all the time, Player 2 will recognize this and play aggressively as well to maximize his payoff (P2 EV = 0.5). Similarly, if player 2 always plays aggressively, Player 1 will begin to play passively to maximize his payoff (P1 EV=1.0). The article summarizes this as follows, “In each cell, one player is better off moving counterclockwise to the next cell, and they chase each other around the matrix [proving this situation has no pure strategy equilibrium].” However, if player 1 now begins to select his strategy at random, the other player can no longer predict his opponent’s strategy; thus, player 2 becomes indifferent to what strategy he should play. The mixed- strategy equilibrium payoffs can be solved for as follows:
Player 2: 0.5q+1(1-q)= 1q+0.5(1-q)
q = 0.5 and EV = 0.25
Player 1: 0.5p+0(1-p)= 0p+0.5(1-p)
p = 0.5 and EV = 0.75
Thus, player 1’s decision to randomize his strategies (i.e. when he decides to bluff) results in him earning a 50% more expected value (EV) than if he didn’t randomize his strategy as before.
The article goes on to generalize this result for larger bets and money pots, which displays an interesting trend in strategies for both players. When the bet size is small compared to the pot, it is always worth it for player 2 to call on a small bet. This is because player 2 then gets a chance at winning a big pot and making sure player 1 is honest. As bets become large compared to the pot, however, it becomes more costly for player 2 to call on these big bets. This is because player 2 must then bet a large amount of money to win a small pot and keep player 1 honest. Therefore, the bluff frequency of player 1 goes up as his bet size increases relative to the pot (see attached image)!
The reader should, however, be aware of the caveats of this analysis. The first and foremost being that most poker players do not truly randomize their strategy, and thus, this analysis could lead to a lower than expected payoff. Additionally, it neglects the more complicated situations where more players are involved and where those players can raise the bet rather than just call or fold. Finally, it assumes player 1 knows he has the better hand. In real poker, players can suspect with a high probability that his hand is superior depending on the actual strength of his cards, but he does not know for sure unless he has a royal flush.
Either way, it would be very neat to test this theory with experiment by recording the number of times players were bluffing when making large bets in a poker match.
Source: http://www.huffingtonpost.com/druce-vertes-cfa/the-mathematics-of-bluffi_b_7989900.html