Question

Suppose you know the following for a particular three-player game: The space of strategy profiles \(S\) is finite. Also, for every \(s \in S\), it is the case that \(u_{2}(s)=3 u_{1}(s), u_{3}(s)=\left[u_{1}(s)\right]^{2}\), and \(u_{1}(s) \in[0,1] .\) (a) Must this game have a Nash equilibrium? Explain your answer. (b) Must this game have an efficient Nash equilibrium? Explain your answer. (c) Suppose that in addition to the information given above, you know that \(s^{*}\) is a Nash equilibrium of the game. Must \(s^{*}\) be an efficient strategy profile? Explain your answer; if you answer "no," then provide a counterexample.

Short answer

(a) Yes, a Nash equilibrium must exist. (b) No, it need not be efficient. (c) No, a counterexample shows it might not be efficient.

Step-by-step solution

Analyze Nash Equilibrium Existence

To determine if a finite game must have a Nash equilibrium, remember that according to Nash's theorem, every finite game (finite set of strategy profiles) has at least one Nash equilibrium. Since the set of strategies is finite in this game, there must exist at least one Nash equilibrium.

Define Efficient Nash Equilibrium

A Nash equilibrium is efficient if it maximizes the total utility across all players. Here, total utility can be represented as \(U(s) = u_1(s) + u_2(s) + u_3(s)\). Substitute the given utility functions: \(U(s) = u_1(s) + 3u_1(s) + [u_1(s)]^2\). Simplifying, \(U(s) = 4u_1(s) + [u_1(s)]^2\). An efficient Nash equilibrium maximizes this function over possible strategy profiles.

Evaluate Need for Efficient Nash Equilibrium

While a Nash equilibrium must exist, it doesn't necessarily have to be efficient unless it's proven that maximizing \(4u_1(s) + [u_1(s)]^2\) corresponds to the Nash equilibrium conditions for all players. Hence, efficient Nash equilibria are not guaranteed unless specific conditions are satisfied.

Consider Specific Nash Equilibrium Efficiency

Given that \(s^*\) is a Nash equilibrium, it may or may not be efficient. Consider \(u_1(s) = 0\) which makes utilities all zero. It could be a Nash equilibrium, but certainly isn't efficient, as choosing \(u_1(s) = 1\) results in \(U(s) = 5\). Thus, \(s^*\) need not be efficient; a counterexample using \(u_1(s) = 0\) confirms this.

Key concepts

Nash Equilibrium

A Nash Equilibrium represents a situation in a game where no player can benefit by changing their strategy while the other players keep theirs unchanged. This concept forms the backbone of game theory.
In a finite game, like the one described in the exercise, Nash's theorem guarantees the existence of at least one Nash equilibrium. This is because the finite nature of the strategy profiles ensures that the players will reach an outcome where no one has anything to gain by switching strategies.
In our exercise, the set of strategies is finite, and therefore, according to Nash's theorem, there is at least one Nash equilibrium. This means that the players can find a steady state where none wants to change their behavior, providing a critical insight into strategic decision-making.

Efficient Strategy

An efficient strategy, or efficient Nash equilibrium, occurs when the sum of utilities across all players is maximized. This makes the strategy beneficial for everyone involved, as it results in the highest possible overall benefit.
To determine efficiency, we calculate the total utility: \(U(s) = u_1(s) + u_2(s) + u_3(s)\). For the given exercise, substituting the utility functions gives \(U(s) = 4u_1(s) + [u_1(s)]^2\).
A strategy profile is efficient if it maximizes this utility function. However, not all Nash equilibria are efficient. Even if a Nash equilibrium is achieved, it doesn't necessarily mean total utility is maximized. Hence, an additional evaluation is required to ensure efficiency.

Utility Maximization

The concept of utility maximization is central to understanding players' decisions in a game. Each player aims to choose a strategy that maximizes their utility function, given the choices of other players.
In our example, the player's utilities are functionally linked: \(u_2(s) = 3u_1(s)\) and \(u_3(s) = [u_1(s)]^2\). This implies that as player 1's utility changes, the others follow predictably.
Utility maximization involves selecting strategy profiles where players reach an agreement that improves or maximizes their earnings or satisfaction. While a Nash equilibrium will ensure players are satisfied with their choices, utility maximization goes further by aiming for the most beneficial outcome for all, aligning with efficient strategies.

Finite Games

Finite games are types of games where the number of players, strategies, and possible outcomes is limited. This characteristic makes them particularly interesting in game theory analysis because they are generally simpler to analyze.
In the exercise, the strategy space \(S\) is finite. This simplifies the problem, as we can apply Nash's theorem to ascertain that at least one Nash equilibrium exists.
The finite nature also means that we can enumerate all possible strategies and outcomes, providing an organized view of the strategic landscape. Finite games can often be represented in matrices or trees, making them accessible and straightforward to understand, allowing deeper insights into the strategic interactions between players.