## This game is over

31 MayIn game theory, the **Nash equilibrium** is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally.If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

### Informal definition

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks himself or herself: “Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?”

If any player would answer “Yes”, then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium.

The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because it may happen that a Nash equilibrium is not Pareto optimal.

The Nash equilibrium may also have non-rational consequences in sequential games because players may “threaten” each other with non-rational moves. For such games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis.

### Formal definition

Let *(S, f)* be a game with *n* players, where *S _{i}* is the strategy set for player

*i*,

*S=S*is the set of strategy profiles and

_{1}× S_{2}… × S_{n}*f=(f*is the payoff function for

_{1}(x), …, f_{n}(x))*x*∈

*S*. Let

*x*be a strategy profile of player

_{i}*i*and

*x*be a strategy profile of all players except for player

_{-i}*i*. When each player

*i*∈{1, …, n} chooses strategy

*x*resulting in strategy profile

_{i}*x = (x*then player

_{1}, …, x_{n})*i*obtains payoff

*f*. Note that the payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player

_{i}(x)*i*as well as the strategies chosen by all the other players. A strategy profile

*x*

^{*}*∈ S*is a Nash equilibrium (NE) 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 game can have either a pure-strategy or a mixed Nash Equilibrium, (in the latter a pure strategy is chosen stochastically with a fixed frequency).

### Nash’s Existence Theorem

Nash proved that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium.

From the Free Online Encyclopedia

## Left or Right Handed

30 MaySome people make a big distinction between left and right handedness.

I’ve been told that most people are right handed. Some of the left handed people I’ve known were snobby about it.

Apparently I am naturally left handed but while in kindergarten when there was still a question about which hand I preferred, Demeter my mother made sure that I used my right hand to write with because here we read and write left to write and who wants to drag their hand across a page filled with ink and smear it. Anyway even in kindergarten when I was mandated to write with my right hand, I was unable to coordinate my brain and right hand with the scissors we used which were right handed scissors to cut out shapes. I used my left hand. Consequently my initial cuts were jagged and not that good at all.