How many mixed strategy nash equilibrium

Once the mixed equilibrium expected utilities are written only in terms of each player own utility payoffs, we can study the variation of the mixed equilibrium expected utility with respect to changes in a given utility payoff through a first-order derivative analysis.

We restrict our attention to marginal changes in a single utility payoff of the game, i. There are two main ideas behind marginal changes: the first one is that the nature of the equilibrium set remains the same, i. These assumptions are needed because we use the payoff order to characterize the games that provide incentives to burning money behavior. Group Decision and Negotiation, 3: Thus, for player 1, we have:. Through these general expressions, we can evaluate how players' mixed equilibrium expected utilities vary, when their respective utility payoffs change, analyzing the sign of the derivatives in Equations 2.

In this section, we discuss necessary and sufficient conditions that guarantee that the derivative of some player expected utility with respect to one of his utility payoff is negative or at least non-positive. Before stating this Lemma, let us present a definition. Definition 1. We say that player 1 resp. Lemma 1. Similarly, in both scenarios for player 2, one can show that there is a unique non-degenerated strategy for player 1 that makes player 2 indifferent between L and R.

Thus, there is a unique mixed Nash equilibrium in the non-degenerated sense. An analogous reasoning is applied for player 1 and therefore will be omitted. The only way to do that is to play a mixed strategy where. Thus, in order for q belong to the interval 0, 1 , we must have that. In both cases, neither player 1 is indifferent between his strategies nor he has a weakly or strongly dominant strategy. Now analyzing the conditions that guarantee non-positive derivatives and strictly negative derivatives, it will be shown that the negative derivatives occurred only in games in which players have a preference that the other uses a particular strategy, regardless of his own choice.

For more details see the original paper. In this case we say that s j weakly collaboratively dominates s'j for player i. Theorems 1 and 2 show that, in games with a unique mixed equilibrium in the non-degenereted sense, a non-positive respectively, negative derivative of player i's mixed equilibrium expected utility with respect to his own utility payoffs occurs if, and only if, player j has a strategy that is weakly respectively, strongly collaboratively dominant for him, player i. Moreover, nonpositive respectively, negative derivatives always occur when are taken with respect to player i's utility payoffs associated with the strategy that is the best response to the weakly resp.

Theorem 1. Thus there are two derivatives of player i's mixed equilibrium expected utility taken with respect to one of his utility payoffs that are non-positive and two that are positive if, and only if, player j has a weakly collaboratively dominant strategy for player i.

Moreover, the non-positive derivatives are always with respect to player i's utility payoffs associated with the strategy that is the best response to the weakly collaboratively dominant strategy of player j and those are the highest and lowest utility payoffs of player i in the game.

Let us consider case A. It follows that , so we should consider the following three sub-cases: , , and. Furthermore, strategy U of player 1 is the best response to L and a is the highest payoff and b is the lowest , while the other derivatives are positive.

Furthermore, strategy D of player 1 is the best response to R and d is the highest payoff and c is the lowest , while the other derivatives are positive.

Theorem 2. Thus there are two derivatives of player i's mixed equilibrium expected utility taken with respect to one of his utility payoffs that are negative and two that are positive if, and only if, player j has a strongly collaboratively dominant strategy for player i. Moreover, the negative derivatives are always with respect to player i's utility payoffs associated with the strategy that is the best response to the strongly collaboratively dominant strategy of player j and those are the highest and lowest utility payoffs of player i in the game.

Suppose that we are in case A. It follows that and. Therefore, consider the following three sub-cases: , ,. In this case, there are no strongly collaboratively dominant strategies and all derivatives are non-negative. In the previous section, we showed that whenever negative derivatives of some player mixed equilibrium expected utility happen, they are with respect to the highest and lowest payoffs of such player. In this section, we answer the following question: assuming that players will play according to the mixed equilibrium, and there are two negative derivatives for a given player, if this player has the opportunity to reduce x units from a given strategy profile, then what is the best utility reducing strategy that the player can adopt?

We prove that he should reduce utility in the case that he uses a strategy that is a best response to the strategy of the other player that is strongly collaboratively dominant for him. However, as we show next, for some cases the player should only reduce utility if the other player i ndeed chooses the strongly collaboratively dominant strategy for him this situation corresponds to reduce utility in his highest utility payoff in the game , while in other cases the opposite should happen this situation corresponds to reduce utility in his lowest utility payoff in the game.

In order to show those results, we should look initially at how the mixed equilibrium strategy of a given player reacts to changes in the payoffs of the other player. Therefore, for player 2, using Equation 2. Now, we can rewrite Equations 2. From these latter equations, it can be seen that the derivative of the expected utility of player 1 is a function of the derivative of player 2's mixed equilibrium strategy. Let us consider Case 1. In this case, strategy R of player 2 is strongly collaboratively dominant for player 1.

Thus, it follows that the derivative of the expected utility of player 1 is negative with respect to payoffs d and c , and R , which, in this case, is strongly collaboratively dominant for player 1.

The analyses of the remaining cases are analogous. Therefore, player 1 should reduce utility with respect to the payoff that provide the greater increase in the probability of player 2 choosing the strategy that is strongly collaboratively dominant for him, player 1.

To emphasize this conclusion, let us analyze the same problem from another perspective. Then, assuming that the general ordering of payoffs in the game is maintained, with respect to what payoff should he reduce these x units of utility?

In addition, it has a mixed strategy. Suppose that Column swerves with probability p. This means that the mixed strategy equilibrium is, in some sense, the more reasonable equilibrium. The nature of the payoffs is that paper beats rock, rock beats scissors, and scissors beats paper. This game has the structure that is illustrated in Figure Previous Section.

Table of Contents. Next Section. Key Takeaways A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy. A Nash equilibrium without randomization is called a pure strategy Nash equilibrium.

If a player is supposed to randomize over two strategies, then both must produce the same expected payoff. The matching pennies game has a mixed strategy and no pure strategy. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.

Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Related Hot Network Questions. Question feed. Therefore he should try to be unpredictale, for as soon as his opponent is able to predict his actions he will be able to take advantage of the situation.

Similarly player 2 must be unpredictable in order to avoid losing while playing this game. A mixed strategy for player i is a probability distribution over his set of available actions. Mixed Strategy Equilibrium In many games players choose unique actions from the set of available actions. Definition: A mixed strategy for player i is a probability distribution over his set of available actions.