Question

Compute the Nash equilibria of the following location game. There are two people who simultaneously select numbers between zero and one. Suppose player 1 chooses \(s_{1}\) and player 2 chooses \(s_{2}\). If \(s_{i}

Short answer

The Nash equilibria are when both players choose the same number, \(s_1 = s_2\), for any \(x\) between 0 and 1.

Step-by-step solution

Understand the Game Structure

The game involves two players who choose numbers between 0 and 1. Each player's payoff depends on the choices made. If one player's choice is less than the other's, they share the sum of their choices as their payoffs. If the choices are equal, they get a payoff of 1/2.

Analyze Case 1 - Equal Choices

When player 1 chooses \(s_1\) and player 2 chooses \(s_2\) such that \(s_1 = s_2\), both players receive a payoff of \(1/2\). This outcome is always a Nash equilibrium because none of the players can unilaterally improve their payoff by changing their choice.

Analyze Case 2 - Unequal Choices

Consider the situation where \(s_1 eq s_2\). If \(s_1 < s_2\), player 1 gets a payoff of \((s_1+s_2)/2\) and player 2 gets \(1-(s_1+s_2)/2\). The players will only be in equilibrium if neither can increase their payoff by choosing a number closer to the other's choice.

Identify Conditions for Nash Equilibrium

For unequal choices to be a Nash equilibrium, we must check if either player can improve their payoff by moving closer to the other's decision. Suppose player 1 chooses \(s_1 < s_2\). If player 1 moves closer to \(s_2\) or matches it, they could potentially improve their payoff.

Conclusion - Nash Equilibria Identification

Upon evaluating both cases, the Nash equilibria occur at points where both players choose identical strategies, \(s_1 = s_2 = x\) for any \(x\) in the interval [0, 1]. This is because any unilateral deviation does not yield better payoffs than 1/2.

Key concepts

Game Theory

Game Theory is an area in mathematics and economics that studies strategic interactions among rational decision-makers. It explores how individuals make choices when their outcomes depend not only on their own actions but also on the actions of others.

To understand game theory, it helps to consider each "game" as a model of real-life scenarios. Here, players (individuals, firms, nations) choose strategies to maximize their benefits or payoffs. These strategies are often chosen simultaneously, and the success depends on the strategies chosen by others.

In the context of a Nash Equilibrium—a core concept in game theory—each player's strategy is optimal given the strategy of the other player. No one can benefit by changing their strategy unilaterally while all others keep theirs unchanged.

Location Game

The Location Game is a special type of game in game theory where players choose locations along a "line" or "interval." The goal is typically to optimize their payoff, taking into account the position choices of others.

In our scenario, the location game involves two players selecting numbers between 0 and 1. The results depend on how their chosen numbers compare. It's an engaging exercise in strategy and forward-thinking because each player's payoff reflects both their position and that of their opponent.

This game's interesting feature is how players must balance their choice: being too aggressive or too conservative can disadvantage them if it doesn't anticipate the rival's position accurately. The challenge is understanding the strategic implications of every possible number they can pick.

Strategy

In any game, the strategy is the plan or method a player uses to reach their objectives. This plan considers the rules of the game and the likely responses of other players.

For the location game, each player's strategy is a choice of a number within the designated range (from 0 to 1). The chosen number represents a "position," optimizing which requires insight into the rival's likely choices.

The best strategy, in Nash Equilibrium terms, is one where neither player can profit by changing their number alone. This makes equal choices an optimal strategy because - If one player deviates, they don't improve since their payoff of 1/2 isn't exceeded.

Understanding player strategy in this framework helps anticipate moves and counter-moves, essential for mastering more complex strategic games.

Payoff Matrix

A payoff matrix is a useful tool in game theory to represent the outcomes for each combination of strategies that players choose. It helps clarify how different strategic choices lead to different results.

In the location game example, although not explicitly laid out, one can imagine a conceptual payoff matrix where:

• If both players select the same number: they each receive a payoff of 1/2.
  
• If player 1 chooses a smaller number than player 2, then their payoff is \((s_1 + s_2)/2\)
  
• Player 2's payoff becomes \((1 - (s_1 + s_2)/2)\)
  

The payoff matrix allows players to visualize potential outcomes clearly, assisting them in deciding optimal strategies. With this visualization, both can assess which situations would likely give them the best results based on their actions and those of their opponent.