Many situations do, in fact, involve several decision makers who compete with one another to arrive at the best outcome. These types of competitive decision-making situations are the subject of game theory. Anyone who has played such games as card games or board games is familiar with situations in which competing participants develop plans of action in order to win. Game theory encompasses similar situations in which competing decision makers develop plans of action in order to win.
Types of Game Situations
Competitive game situations can be subdivided into several categories. One classification is based on the number of competitive decision makers, called players, involved in the game. A game situation consisting of two players is referred to as two-person game. When there are more than two players, the game situation is known as an n-person game.
Games are also classified according to their outcomes in terms of each player's gains and losses. If the sum of the players' gains and losses equals zero, the game is referred to as zero-sum game. In a two-person game, one player's gains represent at another's losses. For example, if one player wins 100, then the other player loses 100; the two values sum to zero. Alternatively, if the sum of the players' gains and losses does not equal zero, the game is known as a non-zero sum game.
The two-person, zero-sum game is the one most frequently used to demonstrate the principles of game theory because it is the simplest mathetimatically.
Example of competitive situations that can be organized into two-person, zero-sum games include
1. a union negotiating a new contract with management
2. two armies participating in a war game
3. two politicians in conflict over a proposed legislative bill, one attempting to secure its passage and the other attempting to defeat it
4. a retail firm trying to increase its market share with a new product and a competitor attempting to minimize the firm's gains
5. a contractor negotiating with a government agent for a contract on a project
Game Strategies
A strategy is a plan of action to be followed by a player. Each player in a game has two or more strategies, only one of which is selected for each playing of a game.
1. A Pure Strategy when each player in the game adopts a single strategy as an optimal strategy, then the game is a pure strategy game. The value of a pure strategy game is the same for both the offensive player and the defensive player. In contrast, in a mixed strategy game, the players adopt a mixture of strategies if the game is played many times.
A pure strategy game can be solved according to the minimax decision criterion. According to this principle, each player plays the game in order to minimize the maximum possible losses. The offensive player will select the strategy with the largest of the minimum payoffs (called the maximin strategy), and the defensive player will select the strategy with the smallest of the maximum payoffs (called the minimax strategy).
2. Dominant Strategies. Dominance occurs when all the payoffs for one strategy are better than the corresponding payoffs for another strategy.
3. A mixed strategy occurs when each player selects an optimal strategy and they do not result in an equilibrium point( the same outcome) when the maximin and mnimax decision criteria are applied.
Practice exercises
1. Payoff Table for Camera Companies
Camera Company 1 Strategies Camera Company 2 Strategies
A B C
1 9 7 2
2 11 8 4
3 4 1 7
2. Two fast foods chains, MacBurger and Burger Doodle, dominate the fast food market. MacBurger is currently the market leader, and Burger Doodle has developed three marketing strategies, encompassing advertising and new product lines, to gain a percentage of the market now belonging to MacBurger. The following payoff table shows the gains for Burger Doodle and the losses for MacBurger given the strategies of each company.
Burger Doodle Strategies MacBurger Strategies
A B C
1 4 3 6
2 -2 5 1
3 3 2 4
Determine the mixed strategy for each company and the expected market share gains for Burger Doodle and losses for Macburger.
3. Consider the following payoff table for a mixed strategy game between two players
Player 1 Strategies Player 2 Strategies
A B C
1 50 60 30
2 10 32 25
3 20 55 4
Determine the mixed strategy for each player and the expected gains and losses that result.
4. Given the following payoff table for a mixed strategy game between two players, determine the strategy and the gains and losses for each player.
Player 1 Strategies Player 2 Strategies
A B C D
1 40 30 20 80
2 90 50 60 65
3 80 75 52 90
4 60 40 35 50
5. Consider the following payoff table for two game players.
Player 1 Strategies Player 2 Strategies
A B C D
1 6 25 18 10
2 12 14 19 11
3 20 15 7 9
4 15 30 21 16
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