By: James McNamara & Anmol Srivats
Nash equilibrium is defined as a state in a multi-player game where every player knows every other player’s strategy and no player has any incentive to change his or her own strategy. This concept may seem purely academic, since it explains something that we already understand, but in addition to a way of formalising the intuitions we all have about human behaviour inside economic theory, the concept has also been responsible for changes in the way we think about the real world.
Applying this concept to real world cases makes it tangible. For example, in the game of poker there are real profits to be made by simply following strategies that minimize the deviation in your own play from ‘perfect’ Nash equilibrium. To push that out of the realms of abstract game theory, think of this strategy as exploiting someone by attacking their weaknesses, and when opponents’ skills increase to the point that they no longer have weaknesses, your best option is to remove your own weaknesses so as to not be exploitable.
Within online poker (which removes elements such as tells), there are two ways to denominate every strategy – exploitable and exploitative. Exploitable strategies are ones that deviate from the Nash equilibrium, while exploitative strategies deviate from Nash specifically to exploit opponents’ exploitable strategies. By definition, exploitative strategies must always deviate less from Nash than exploitable strategies, otherwise they would be exploitable strategies. Therefore under certain conditions, the optimal strategy is more about not being exploited than about exploiting the weaknesses of others. Closely following Nash equilibrium becomes a fairly strong exploiting strategy, and by definition, following ‘perfect’ Nash is a non-exploitable strategy.
A trivial application of this reasoning can be seen in bluffs and calls. If another player bluffs very often when he bets, you will call very often. Suppose that some reason – perhaps he is trying to impress a lady friend – he knows that you will call, but still perceives it to be in his best interest to continue his strategy. Thus neither of you would opt to change your strategy: he would continue to bluff, and you would continue to call. Since the strategies don’t change, they are in a static state – i.e., they are in equilibrium.
The strategies sound simple enough, but the calculation of Nash equilibrium is not. When two people play a hand of online poker, the number of possible combinations of cards that can be dealt equals 5.56*10^13. To obtain Nash equilibrium, you need to compute each of these combinations as well as decide things such as the optimal bet size on each of four betting rounds. Altogether, these decisions would push the exponent well beyond 10^20. In addition, you may be playing games with six or nine players at your table, which would increase the number of card combinations to 2*10^25 and 3*10^33 respectively. Because the number of card combinations and the number of betting combinations are so high, no computer will find the Nash equilibrium of poker any time soon. Thus, the best players in the world have to estimate the Nash equilibrium, and compete between their estimations.
Interestingly, online variations of the standard poker format, such as ‘Hyper-Turbo’ tables, create a more rigid structure of the game in which Nash strategy is not only easier to follow but, with practice, can reward clever players. The game starts with four players each holding a stack consisting of 10 big blinds (as opposed to regular games, which are 100 big blinds deep), and the blinds double every 3 minutes, meaning that a typical tournament lasts only 5 minutes. Due to the short stack sizes, there is not much play after the flop in the games, and the vast majority of players’ decisions are either ‘all in’ or ‘fold’ before the flop is dealt. Since we have a good idea of the probability of two hands beating each other in a pre-flop scenario, and since decisions are binary (all in or fold), it is easier to estimate a Nash equilibrium for these games than for standard games. In addition, because the Nash equilibrium is relatively easy to calculate, most players are not playing particularly exploitable strategies so your edge is relatively small resulting in a large variance. However, since you will play a large number of games, this variance becomes unimportant. Over time, the strength of your conformity to Nash will ultimately determine your success and earning potential.
Since the number of decisions is limited to two for all players, the card combinations are limited to one of 169 hands (suited, off-suit, and pairs), and generally only two players go all in against each other at once, a Nash equilibrium estimate requires going through a ‘mere’ 600,000 combinations, which a computer can do reasonably quickly. Thus it is possible to study and better yourself at this variation of poker with the help of software, which is easily available by Googling ‘ICM Nash calculator’.
Successful application of this simple economic concept – combined with dedication and a sliver of luck – can be profitable, especially on higher stakes tables where both risks and rewards escalate.