Examples of Games Without Nash Equilibrium in Game Theory
Understanding the concept of Nash equilibrium in game theory provides insights into strategic decision-making under the assumption of rationality. However, not all games maintain a Nash equilibrium. This article explores the examples of games without a Nash equilibrium, focusing on games with infinite strategy spaces and trivial counterexamples. We will delve into the intricacies of game theory and provide detailed explanations to ensure clarity.
Introduction to Nash Equilibrium
Nash equilibrium is a fundamental concept in game theory, named after John Forbes Nash Jr. It refers to a stable state of a game where no player can benefit by deviating from their chosen strategy unilaterally, assuming the other players' strategies remain unchanged. In simpler terms, each player is playing their best response to the strategies of the other players.
Common Example: Rock Paper Scissors
One of the most familiar examples demonstrating the absence of a Nash equilibrium is the game of rock paper scissors. If one player rigidly follows a deterministic strategy, such as always playing rock, the opponent can easily counter by always playing paper, leading to a predictable outcome. The best strategy is to play randomly with each option, turning the game into a mixed strategy where each choice has an equal probability of 1/3. In this scenario, no player can gain an advantage by changing their strategy unilaterally.
Trivial Counterexample with Infinite Strategy Space
When it comes to games with infinite strategy spaces, there are numerous examples where a pure strategy Nash equilibrium does not exist. The following counterexample is a straightforward illustration:
Game Setup: There are two players. Each player picks a real number between 0 and 1, inclusive. Payoff Rules: If both players pick the same number, they both get a payoff of 0. Otherwise, each player gets the number they picked as a payoff.This game lacks any pure strategy Nash equilibrium. If one player chooses 1, the best response for the other player would be to choose the largest number less than 1, but since 1 is not well-defined, the other player cannot perfectly adapt their strategy. Conversely, if neither player picks 1, one player can improve their outcome by deviating to playing 1. This creates a continuous loop of potential deviations and counter-strategies.
Beyond Pure Strategies: Mixed Strategies
The absence of pure strategy Nash equilibria can also extend to mixed strategies. In the game described above, if one player decides to play 1 with some probability, the other player must adapt their strategy accordingly. The issue arises when one player plays some combination of numbers, as demonstrated in the following example:
Assume player A plays 1 with probability ( p ). Then, the best response for player B is to play a mixed strategy consisting of numbers close to 1. However, player A can always find an ( epsilon > 0 ) such that player B is not playing ( 1 - epsilon ) with positive probability. By doing so, player A can guarantee a payoff of ( 1 - epsilon times p ), leading to an infinitesimal advantage over player B.
Conclusion
Games without Nash equilibria are an intriguing aspect of game theory, particularly those involving infinite strategy spaces. While rock paper scissors provides a familiar example, more complex games such as the one described with infinite real number choices illustrate the underlying principles of strategic uncertainty and the absence of stable equilibrium points.