Game theory is a branch of mathematics that analyzes the behavior of individuals or groups in decision-making situations. In many cases, it is essential to identify the solution concept that governs the game being played.

One such solution concept is Socially Optimal Equilibrium (SPE), which represents an outcome where all players receive their maximum possible payoffs. In this article, we will discuss how to find the SPE in game theory.

## What is SPE?

Before we dive into finding the SPE, it is crucial to understand what it means. Socially Optimal Equilibrium (SPE) is a solution concept in game theory, which represents a set of strategies that result in the highest possible payoff for each player, given the other players’ strategies.

The SPE is considered socially optimal because it maximizes the sum of payoffs received by all players and leads to an efficient outcome. It provides a benchmark for evaluating other non-cooperative equilibrium concepts such as Nash equilibrium.

## How to Find SPE?

Finding SPE requires analyzing all possible strategies and outcomes for each player. It can be done by following these simple steps:

### Step 1: Identify the Strategy Set

The first step is to identify all possible strategies available to each player in the game. A strategy set represents all possible actions that a player can take while playing a game. For example, if two players are playing rock-paper-scissors, their strategy set would be {rock, paper, scissors}.

### Step 2: Construct Payoff Matrix

The next step is to construct a payoff matrix that represents all possible outcomes based on different combinations of strategies chosen by each player. A payoff matrix shows how much each player will receive as a reward or penalty for every combination of strategies they choose.

For instance, let’s consider a simple two-player game where Player 1 and Player 2 can either cooperate or defect. The payoff matrix for this game may look like:

Cooperate | Defect | |
---|---|---|

Cooperate | (2,2) | (0,4) |

Defect | (4,0) | (1,1) |

In the above table, the first number in each cell represents Player 1’s payoff, and the second number represents Player 2’s payoff.

### Step 3: Check for Dominant Strategies

A dominant strategy is a strategy that provides a player with a higher payoff regardless of the other player’s strategy. In other words, it is always the best choice for a player to adopt a dominant strategy.

To find dominant strategies in a game, we need to compare each player’s payoffs for every combination of strategies. If one strategy dominates all others for a particular player, then that is their dominant strategy.

In our example game, neither player has a dominant strategy since there isn’t any strategy that dominates all others.

### Step 4: Identify Nash Equilibrium (NE)

A Nash equilibrium (NE) is an outcome where no player can increase their payoff by changing their strategy unilaterally. In other words, given the strategies chosen by other players, each player is playing their best response to those strategies.

To identify NE in a game, we need to check if there exists any combination of strategies where each player’s strategy is their best response to the other players’ strategies. If such an outcome exists, it is a Nash equilibrium.

In our example game, (Defect, Defect) is the only Nash equilibrium since both players’ strategy is their best response to the other player’s strategy.

### Step 5: Determine SPE

Once we have identified all the Nash equilibria in a game, we need to check which one of them results in the highest possible payoff for each player. The Nash equilibrium that maximizes the sum of payoffs received by all players is known as Socially Optimal Equilibrium (SPE).

In our example game, (Cooperate, Cooperate) results in the highest possible payoff for each player and is, therefore, the SPE.

## Conclusion

Socially Optimal Equilibrium (SPE) represents an outcome where all players receive their maximum possible payoffs. To find SPE in a game, we need to identify all possible strategies and outcomes for each player and determine which combination of strategies results in socially optimal outcomes. By following the above steps, one can easily find SPE in any game theory problem.