Game theory was introduced in the previous chapter to better understand oligopoly. Recall the definition of game theory.

Game Theory = A framework to study strategic interactions between players, firms, or nations.

Game theory is the study of strategic interactions between players. The key to understanding strategic decision making is to understand your opponent’s point of view, and to deduce his or her likely responses to your actions.

A game is defined as:

Game = A situation in which firms make strategic decisions that take into account each other’s’ actions and responses.

A payoff is the outcome of a game that depends of the selected strategies of the players.

Payoff = The value associated with a possible outcome of a game.

Strategy = A rule or plan of action for playing a game.

An optimal strategy is one that provides the best payoff for a player in a game.

Optimal Strategy = A strategy that maximizes a player’s expected payoff.

Games are of two types: cooperative and noncooperative games.

Cooperative Game = A game in which participants can negotiate binding contracts that allow them to plan joint strategies.

Noncooperative Game = A game in which negotiation and enforcement of binding contracts are not possible.

In noncooperative games, individual players take actions, and the outcome of the game is described by the action taken by each player, along with the payoff that each player achieves. Cooperative games are different. The outcome of a cooperative game will be specified by which group of players become a cooperative group, and the joint action that the group takes. The groups of players are called, “coalitions.” Examples of noncooperative games include checkers, the prisoner’s dilemma, and most business situations where there is competition for a payoff. An example of a cooperative game is a joint venture of several companies who band together to form a group (collusioin).

The discussion of the prisoner’s dilemma led to one solution to games: the equilibrium in dominant strategies. There are several different strategies and solutions for games, including:

Dominant strategy

Nash equilibrium

Maximin strategy (safety first, or secure strategy)

Cooperative strategy (collusion).

Equilibrium in Dominant Strategies

The dominant strategy was introduced in the previous chapter.

Dominant Strategy = A strategy that results in the highest payoff to a player regardless of the opponent’s action.

Equilibrium in Dominant Strategies = An outcome of a game in which each firm is doing the best that it can regardless of what its competitor is doing

Recall the prisoner’s dilemma from Chapter Five.

Prisoner’s Dilemma: Dominant Strategy

(1) If \(\text{B CONF, A}\) should \(\text{CONF } (8 < 15)\)

(2) If \(\text{B NOT, A}\) should \(\text{CONF } (1 < 3)\)

…\(A\) has the same strategy \(\text{(CONF)}\) no matter what \(B\) does.

(3) If \(\text{A CONF, B}\) should \(\text{CONF } (8 < 15)\)

(4) If \(\text{A NOT, B}\) should \(\text{CONF } (1 < 3)\)

…\(B\) has the same strategy \(\text{(CONF)}\) no matter what \(A\) does.

Thus, the equilibrium in dominant strategies for this game is \(\text{(CONF, CONF) } = (8,8)\).

Nash Equilibrium

A second solution to games is a Nash Equilibrium.

Nash Equilibrium = A set of strategies in which each player has chosen its best strategy given the strategy of its rivals.

To solve for a Nash Equilibrium:

(1) Check each outcome of a game to see if any player wants to change strategies, given the strategy of its rival.

(a) If no player wants to change, the outcome is a Nash Equilibrium.

(b) If one or more player wants to change, the outcome is not a Nash Equilibrium.

A game may have zero, one, or more than one Nash Equilibria. The Prisoner’s Dilemma is shown in Figure \(\PageIndex{1}\). We will determine if this game has any Nash Equilibria.

Prisoner’s Dilemma - Nash Equilibrium

(1) Outcome \(= \text{ (CONF, CONF)}\)

(a) Is \(\text{CONF}\) best for \(A\) given \(\text{B CONF}\)? Yes.

(b) Is \(\text{CONF}\) best for \(B\) given \(\text{A CONF}\)? Yes.

…\(\text{(CONF, CONF)}\) is a Nash Equilibrium.

(2) Outcome \(= \text{ (CONF, NOT)}\)

(a) Is \(\text{CONF}\) best for \(A\) given \(\text{B NOT}\)? Yes.

(b) Is \(\text{NOT}\) best for \(B\) given \(\text{A CONF}\)? No.

…\(\text{(CONF, NOT)}\) is not a Nash Equilibrium.

(3) Outcome \(= \text{ (NOT, CONF)}\)

(a) Is \(\text{NOT}\) best for \(A\) given \(\text{B CONF}\)? No.

(b) Is \(\text{CONF}\) best for \(B\) given \(\text{A NOT}\)? Yes.

…\(\text{(NOT, CONF)}\) is not a Nash Equilibrium.

(4) Outcome \(= \text{ (NOT, NOT)}\)

(a) Is \(\text{NOT}\) best for \(A\) given \(\text{B NOT}\)? No.

(b) Is \(\text{NOT}\) best for \(B\) given \(\text{A NOT}\)? No.

…\(\text{(NOT, NOT)}\) is not a Nash Equilibrium.

Therefore, \(\text{(CONF, CONF)}\) is a Nash Equilibrium, and the only one Nash Equilibrium in the Prisoner’s Dilemma game. Note that in the Prisoner’s Dilemma game, the Equilibrium in Dominant Strategies is also a Nash Equilibrium.

Advertising Game

In this advertising game, two computer software firms (Microsoft and Apple) decide whether to advertise or not. The outcomes depend on their own selected strategy and the strategy of the rival firm, as shown in Figure \(\PageIndex{2}\).

Advertising: Dominant Strategy

(1) If \(\text{APP AD, MIC}\) should \(\text{AD } (20 > 5)\)

(2) If \(\text{APP NOT, MIC}\) should \(\text{NOT } (14 > 10)\)

…different strategies, so no dominant strategy for Microsoft.

(3) If \(\text{MIC AD, APP}\) should \(\text{AD } (20 > 5)\)

(4) If \(\text{MIC NOT, APP}\) should \(\text{NOT } (14 > 10)\)

…different strategies, so no dominant strategy for Apple.

Thus, there are no dominant strategies, and no equilibrium in dominant strategies for this game.

Advertising: Nash Equilibria

(1) Outcome \(= \text{ (AD, AD)}\)

(a) Is \(\text{AD}\) best for \(MIC\) given \(\text{APP AD}\)? Yes.

(b) Is \(\text{AD}\) best for \(\text{APP\) given \(\text{MIC AD}\)? Yes.

…\(\text{(AD, AD)}\) is a Nash Equilibrium.

(2) Outcome \(= \text{ (AD, NOT)}\)

(a) Is \(\text{AD}\) best for \(\text{MIC}\) given \(\text{APP NOT}\)? No.

(b) Is \(\text{NOT}\) best for \(\text{APP}\) given \(\text{MIC AD}\)? No.

…\(\text{(AD, NOT)}\) is not a Nash Equilibrium.

(3) Outcome \(= \text{ (NOT, AD)}\)

(a) Is \(\text{NOT}\) best for \(\text{MIC}\) given \(\text{APP AD}\)? No.

(b) Is \(\text{AD}\) best for \(\text{APP}\) given \(\text{MIC NOT}\)? No.

…\(\text{(NOT, AD)}\) is not a Nash Equilibrium.

(4) Outcome \(= \text{ (NOT, NOT)}\)

(a) Is \(\text{NOT}\) best for \(\text{MIC}\) given \(\text{APP NOT}\)? Yes.

(b) Is \(\text{NOT}\) best for \(\text{APP}\) given \(\text{MIC NOT}\)? Yes.

…\(\text{(NOT, NOT)}\) is a Nash Equilibrium.

There are two Nash Equilibria in the Advertising game: \(\text{(AD, AD)}\) and \(\text{(NOT, NOT)}\). Therefore, in the Advertising game, there are two Nash Equilibria, and no Equilibrium in Dominant Strategies.

It can be proven that in game theory, every Equilibrium in Dominant Strategies is a Nash Equilibrium. However, a Nash Equilibrium may or may not be an Equilibrium in Dominant Strategies.

Maximin Strategy (Safety First; Secure Strategy)

A strategy that allows players to avoid the largest losses is the Maximin Strategy.

Maximin Strategy = A strategy that maximizes the minimum payoff for one player.

The maximin, or safety first, strategy can be found by identifying the worst possible outcome for each strategy. Then, choose the strategy where the lowest payoff is the highest.

We use Figure \(\PageIndex{2}\) to find the Maximin Strategy for the Prisoner’s Dilemma.

(1) Player \(A\)

(a) If \(\text{CONF}\), worst payoff \(= 8\) years.

(b) If \(\text{NOT}\), worst payoff \(= 15\) years.

…\(A\)’s Maximin Strategy is \(\text{CONF } (8 < 15)\).

(2) Player \(B\)

(a) If \(\text{CONF}\), worst payoff \(= 8\) years.

(b) If \(\text{NOT}\), worst payoff \(= 15\) years.

…\(B\)’s Maximin Strategy is \(\text{CONF } (8 < 15)\).

Therefore, the Maximin Equilibrium for the Prisoner’s Dilemma is \(\text{(CONF, CONF)}\). This outcome is also an Equilibrium in Dominant Strategies, and a Nash Equilibrium.

Advertising Game: Maximin Strategy (Safety First)

(1) \(\text{MICROSOFT}\)

(a) If \(\text{AD}\), worst payoff \(= 10\).

(b) If \(\text{NOT}\), worst payoff \(= 5\).

…MICROSOFT’s Maximin Strategy is \(\text{AD } (5 < 10)\).

(2) \(\text{APPLE}\)

(a) If \(\text{AD}\), worst payoff \(= 10\).

(b) If \(\text{NOT}\), worst payoff \(= 5\).

…\(\text{APPLE}\)’s Maximin Strategy is \(\text{AD } (5 < 10)\).

Therefore, the Maximin Equilibrium in the Advertising game is \(\text{(AD, AD)}\). Recall that this outcome is one of two Nash Equilibria in the advertising game: \(\text{(AD, AD)}\) and \(\text{(NOT, NOT)}\). If both players choose Maximin, there is only one equilibrium: \(\text{(AD, AD)}\).

The relationships between the game theory strategies can be summarized:

An Equilibrium in Dominant Strategies is always a Maximin Equilibrium.

A Maximin Equilibrium is NOT always an Equilibrium in Dominant Strategies.

An Equilibrium in Dominant Strategies is always a Nash Equilibrium. A Nash Equilibrium is NOT always an Equilibrium in Dominant Strategies.