Soapy Inc. and Suddies Inc., the only soappowder producers,...

Soapy Inc. and Suddies Inc., the only soappowder producers, collude and agree to share the market equally. If neither firm cheats, each makes $\$ 1$ million. If one firm cheats, it makes $\$ 1.5$ million, while the complier incurs a loss of $\$ 0.5$ million. If both cheat, they break even. Neither firm can monitor the other's actions. a. What are the strategies in this game? Construct the payoff matrix for this game. b. If the game is played only once what is the equilibrium? Is it a dominant-strategy equilibrium? Explain.

Soapy Inc. and Suddies Inc., the only soappowder producers, collude and agree to share the market equally. If neither firm cheats, each makes $\$ 1$ million. If one firm cheats, it makes $\$ 1.5$ million, while the complier incurs a loss of $\$ 0.5$ million. If both cheat, they break even. Neither firm can monitor the other's actions.
a. What are the strategies in this game? Construct the payoff matrix for this game.
b. If the game is played only once what is the equilibrium? Is it a dominant-strategy equilibrium? Explain.

Key Concepts

Strategies

In game theory, strategies refer to the complete set of actions a player can choose from in a game. Each player's strategy represents a decision rule or plan that determines their actions under different circumstances. In a simultaneous move game, each player's strategy is chosen without knowing the choices of the other players, which is key to predicting outcomes in competitive scenarios.

Payoff Matrix

A payoff matrix is a table that summarizes the outcomes of a game for all possible combinations of strategies chosen by the players. Each cell of this matrix represents the payoffs for each player when specific strategies are played jointly. This tool helps analyze strategic interactions by clearly presenting the rewards and penalties associated with different moves, which is essential for predicting strategic behavior.

Dominant Strategy

A dominant strategy is one that results in the highest possible payoff for a player, regardless of what the other players do. When a player has a dominant strategy, they can choose it with confidence because it maximizes their gain no matter the opponents' decisions. In many strategic situations, the existence of a dominant strategy simplifies the analysis by reducing the complexity of potential outcomes.

Nash Equilibrium

A Nash equilibrium occurs when each player's strategy is optimal given the strategies chosen by the other players. In this state, no player has an incentive to unilaterally deviate from their chosen strategy because any change would lead to a less favorable outcome. In the context of a one-shot game, Nash equilibrium encapsulates the idea of mutual best responses in a scenario where information about others' choices is incomplete or simultaneous.

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