Nash Poker Equilibrium

In poker, two souls coexist the instinctive one, which brings adrenaline and competitiveness, and the rational one allows complex reasoning and mathematical calculations. 

Are the two sides of the conflict? No, on the contrary. Poker can even strike a balance: let’s see how.


John Nash’s Theory Of Equilibrium

Initially, this concept was used exclusively in the theory of Texas Holdem Poker in its Heds-up variant. However, it has also been re-evaluated in the studies concerning tournaments with more than two players. Let’s talk specifically about the final stages of these tournaments. In fact, at this moment, the knowledge of the other players sitting at the table can be defined as good, and their strategies can be understood.

But what is meant the “concept of poker balance”? 

For the first time in 1950 precisely John Nash, an American mathematician, an economist, but met with little success. According to this theory, at the table, every rational player will implement a strategy that can maximize their earnings.

But when is the balance achieved? When the player who participates in a certain game makes his rational move, thinking that the other players will not change their behavior based on this choice.

In simple terms, equilibrium occurs when a player, enabled to know exactly the opponent’s moves, would not make a move other than the one already decided.

The situation that arises is the following: all the players at the table make a rational choice to optimize their profit, regardless of the choices of others. If each player makes the most logical move for himself, and everyone did as he did, no one would be in a different position from the others, unable to further improve their position. Each would then be bound to the choices of the other.

The advantage? Surely in such a situation, it is very difficult to make “dirty” plays, trash hand or bluff, unless the opponent has calculated an advantage that could really bring enormous benefit only to himself.

The Prisoner’s Dilemma

Two criminals are arrested and divided for interrogation, thus unable to communicate with each other. Everyone can choose whether to confess or not. If only one confesses, then the one who chose to confess will have avoided the sentence and sentenced the other (for example, to 4 years in prison). If they both confess, they will each have a penalty (say 2 years). If neither confesses, both will serve a sentence (a few months).

How would the two prisoners act if they knew these rules? According to the Nash equilibrium, the best choice for both would be to talk, thus avoiding the maximum penalty if one of the two confesses and the other does not. In reality, knowing the rules, we know that the convenient choice would be not to confess: but it is risky since the other’s intentions are not known. Only if there had been a way to agree, the two could have played as a team, but this is not a balance as the strategy of “does not confess” is linked to that of “confess,” and there is no certainty about the penalty for the two prisoners.

What Can We Conclude?

Cooperation leads to maximizing profit: when a player aims with his strategy to improve only his earnings, regardless of the choices of others, we arrive at a stalemate in which it is useless to change strategy to improve one’s position. The improvement occurs only when the players collaborate so that there is again for everyone.

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