# Nash Equilibrium : Deviate from the optimal

Nash Equilibrium defines a simple yet very powerful aspect of interacting entities in a group. It states a solution set of a game involving two or more players where each player is in full knowledge of the equilibrium strategies of all the other players and in which none can gain by changing his strategy unilaterally. Stated simply, each player makes a decision that is best in interest of the group and him in accordance with the decisions of all the other members of the group playing the game. In case of a player changing his strategy unilaterally he finds himself isolated and is left with only a choice of reverting back to the set equilibrium strategy as none of the other members would change their strategies due to lack of improved incentives and the payoffs.

Formally, let *(S, f)* be a game with *n* players, where *Si* is the strategy set for player *i*, *S=S1 X S2 … X Sn* is the set of strategy profiles and *f=(f1(x), …, fn(x))* is the payoff function for *x*

*S*. Let *xi* be a strategy profile of player *i* and *x-i* be a strategy profile of all players except for player *i*. When each player *i*{1, …, n} chooses strategy *xi* resulting in strategy profile *x = (x1, …, xn)* then player *i* obtains payoff *fi(x)*. The payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player *i* as well as the strategies chosen by all the other players. A strategy profile *x*S* is a Nash equilibrium if no unilateral deviation in strategy by any single player is profitable for that player, that is

When the inequality above holds strictly (with > instead of ≥) for all players and all feasible alternative strategies, then the equilibrium is classified as a strict Nash equilibrium. If instead, for some player, there is exact equality between x*i* and some other strategy in the set *S*, then the equilibrium is classified as a weak Nash equilibrium.

**A General Case**

To understand what aspect of the Nash Equilibrium will be dealt with in this work let us consider the following example. Suppose there are two friends X and Y. Both want to spend quality time together. Now the idea of quality time according to X is to go on a movie and for Y it is a stroll in a park. Although none has any problem on the plan of other but their moods and hence their payoffs vary accordingly as shown in figure 1.

The above table assumes arbitrary payoffs signifying the relative happiness of each friend under different conditions.

For X and Y, if both agree upon a movie then the payoff is (2,2).

If X goes for a movie alone and Y for a stroll alone then it is (1,0), i.e. neither of them want to go alone.

Similarly if X goes for a stroll and Y goes for movie alone then it is (0,1) as none wants to go alone.

In the last case both agree upon a stroll in the park which has payoff (1,1), mainly because X does not like strolls and is continuously grumbling which makes Y’s mood go off.

A simple agreement to the situation would be that both agree to go for a movie as it offers high payoffs to both the players and deviating from them unilaterally would not yield a better payoff.

This was one arbitrary solution to the problem. Suppose we have a variation to it as in figure 2.

Suppose it’s the birthday of Y and hence X decides to keep him happy and agrees upon a walk with no grumbling making Y all the more happy with payoff of (1,3).

Now in general cases a payoff of (1,3) would not be considered over a payoff of (2,2). But since here it’s the Birthday of Y, X decides to agree upon the situation as it is a special day for Y.

Considering a computing situation let us assume that movie and stroll are resources and X and Y are processes to which these would be allocated. Then figure 1 would be a general assignment case wherein both processes are equally important. The figure 2 would then be signifying that process Y is a critical process and hence an exception for it has to be applied as the system could not risk upon this process entering a deadlock.

The thing to note here is that the former solution was in accordance with Nash Equilibrium but the later somewhat defies it as X agrees upon a strategy it is getting a lower payoff even in presence of a more favorable strategy for it. This is the general case of exception consideration that has to be incorporated into systems following Nash Equilibrium solutions for better desired results rather than better obtainable results according to Nash Equilibrium.

This can be more understood by critically examining the solutions of problems like Prisioner’s dilemma and Braess’s Paradox.

Another thing to note here is that the latter case in Figure 2 is not a general one, i.e. under normal conditions solution of Figure 1 would hold true which is a more optimal solution. Figure 2’s solution would come into play only when we require the best solution for a particular player and an optimal for the rest.