# How to Using IESDS and Pure Strategy Nash Equilibrium to Solve a Game

Two Methods:Using IESDSPure Nash Equilibrium

Game Theory is an area of Economics where the payoff, profit or happiness is determined by the actions of two or more players in the situation, or game. Being able to predict the outcome of a game is a very critical tool that can be used to predict situations in the real world. Whether it be a business transaction, or simply a situation between two individuals, the most important thing to remember in game theory is the concept of an externality, meaning the choices of someone else will ultimately affect my payoff. Two of the simplest ways to solve a game are finding the Pure Strategy Nash Equilibrium and using the Iterated Elimination of Strictly Dominated Strategies (IESDS). The only materials needed are a payoff matrix. A payoff matrix depicts the players in a game, their strategies in the rows and columns, and each players respective payoffs within the matrix.

## Steps

### Method 1 Using IESDS

- 1
**Identify the players and which strategies are possible for each player.**The first number in each possible outcome of the payoff matrix is the first player's payoff, and the second number is the second player's payoff. - 2
**Identify any strictly dominated strategies.**First look at all the payoffs for player two if they select strategy L and the payoffs if they select strategy R. If*all*of the payoffs in one strategy are higher than the payoffs in the other, we can say the strategy is strictly dominated. - 3
**Eliminate any strictly dominated strategies from the selection.**Any rational player will never select a strategy that gives them a lower payoff.- Strategy R is strictly dominated by strategy L so it may be eliminated from the game, now we look at player 1’s strategies to determine if any strategies are strictly dominated. (We do not need to take into consideration any payoffs that have been eliminated.)
- Strategy D strictly dominates strategy U as 3 is greater than 4 for the first player’s payoff. Therefore we can eliminate strategy U.

- 4
**Determine the remaining strategies.**The only situation left is player one selecting D with a payoff of 4 and player two selecting L with a payoff of 4. We write that the game is solvable with an outcome of (D, L). Note we write the strategies, not the payoff, and the first player’s strategy is written first.

### Method 2 Pure Nash Equilibrium

- 1
**Use IESDS first.**For games not Solvable using IESDS, we will be using the concept of a best response to determine the Pure Strategy Nash Equilibrium of a game.- To begin we will use IESDS to show that this game is not solvable using only IESDS. Using IESDS we can see that for player 2 strategy C is strictly dominated by strategy A and therefore can be eliminated.
- Also using IESDS we can see that strategy H is strictly dominated by strategy E, and can be eliminated.
- We cannot eliminate any more strategies as none are strictly dominated. (Note: equal payoffs does not imply strict dominance.)

- 2
**Once the strictly dominated strategies have been eliminated, select each player's best response to find any pure strategy Nash equilibria.**This is done by looking at each strategy a player can use, and then selecting the other player’s best response to that strategy.- If player two chooses strategy A, the highest payoff player one can achieve is 8 by selecting strategy G. If player one selects strategy E, the highest payoff player two can achieve is 7 by selecting strategy B. Repeat and find all best responses for each strategy. Signify the best response by circling the appropriate payoff.

- 3
**Find the equilibria.**The pure strategy Nash equilibria of the game are the outcomes where both players’ payoffs are circled, indicating solutions to the game. Note that the equilibrium cannot occur with a strictly dominated strategy.

## Warnings

- If game is not solvable using either of these methods, more advanced methods such as Mixed Nash Equilibrium may be necessary. Also note these methods are meant for simple games one stage games, more steps are needed for repeated finite and infinite games.

## Article Info

Categories: Economics