Problem 229 - An Introduction to Game Theory

Question

Consider the following strategy pair in the game for \(K=100 .\) For \(k=1, \ldots, 90\), challenger \(k\) stays out after any history in which every previous challenger that entered was fought (or no challenger entered), and otherwise enters; challengers 91 through 100 enter. The chain-store fights every challenger up to challenger 90 that enters after a history in which it fought every challenger that entered (or no challenger entered), acquiesces to any of these challengers that enters after any other history, and acquiesces to challengers 91 through 100 regardless of the history. Find the players' payoffs in this strategy pair. Show that the strategy pair is not a subgame perfect equilibrium: find a player who can increase her payoff in some subgame. How much can the deviant increase its payoff?

Step-by-Step Solution

Answer

Chain-store fights up to challenger 90, acquiesces to 91-100. Challengers 91-100 can increase payoffs by challenging. Challengers 1-90 can deviate to increase payoffs in some subgames.

Step 1: Understand the Strategy Pair

Identify the strategy pairs given. For challengers 1 to 90, they stay out if all previous challengers were fought or no one entered. For challengers 91 to 100, they always enter. The chain-store fights challengers 1 to 90 if it has fought all previous entering challengers or none entered, and acquiesces in all other situations. For challengers 91 to 100, it always acquiesces.

Step 2: Calculate Payoffs for Challenger 91 to 100

Since challengers 91 to 100 always enter and the chain-store always acquiesces, each of these challengers obtains their entry payoff without any contest.

Step 3: Calculate Payoff for Challenger 1 to 90

For challengers 1 to 90, each will stay out or enter based on the previous challengers' outcomes. The chain-store will fight if it has been fighting all previous entering challengers or if no one has entered. We need to calculate the payoffs considering these conditions.

Step 4: Verify Subgame Perfection

In subgame perfection, each player's strategy should be optimal at every stage of the game. Check if any player (challenger or chain-store) can increase their payoff by deviating from their current strategy in some subgame.

Step 5: Find Potential Deviations

Examine each potential deviation to see if a player can increase their payoff. For example, consider if the chain-store deviates by fighting or acquiescing differently at any point, or if a challenger deviates by choosing to enter or stay out differently. Calculate how much the deviant's payoff could increase.

Key Concepts

Subgame Perfect Equilibrium

Understanding subgame perfect equilibrium (SPE) is crucial for analyzing strategies in game theory. SPE refers to a refinement of Nash Equilibrium, where strategies must form a Nash Equilibrium in every subgame of the original game.
Larger games are divided into smaller subgames, and players must choose strategies that are optimal for every subgame, ensuring no player has an incentive to deviate at any stage.
In our exercise, examining the subgame for each challenger involves assessing if both the challengers and the chain-store choose strategies that maximize their payoffs at each possible point in the game. For example, does the strategy of challengers 1 to 90 always staying out after certain histories produce maximal payoffs?
Ultimately, a strategy pair is subgame perfect if every subgame's outcome stays optimal, reinforcing consistency and optimal behavior throughout the game.

Strategy Pair

A strategy pair defines the actions both players choose in every situation throughout the game.
In our exercise, strategy pairs are specifically described for challengers and the chain-store:

• Challengers 1 to 90 stay out if all previous challengers were fought or no one entered; otherwise, they enter.
• Challengers 91 to 100 always enter.
• The chain-store fights challengers 1 to 90 if it has fought all previous entering challengers or none entered; it acquiesces in all other scenarios.
• For challengers 91 to 100, the chain-store always acquiesces.

Analyzing these strategies helps us break down how both parties act under different circumstances. It’s essential to verify if these predefined strategies lead to a consistent equilibrium where neither the challengers nor the chain-store can improve their outcomes by deviating.

Payoff Calculation

Payoff calculation is essential to understand the benefits players get from following their strategies.
Each player’s payoff depends on the path taken through the game. In our exercise:

• For challengers 91 to 100, since they always enter and the chain-store always acquiesces, their payoffs are straightforward. Each of these challengers secures their entry payoff without a contest.
• For challengers 1 to 90, their decision to stay out or enter depends on the chain-store's historical behavior. If the chain-store fought all previous entering challengers or none entered, any new entry would result in a fight, impacting payoffs negatively. To simplify, each potential history must be evaluated to see if entering or staying out maximizes the challengers’ payoffs.

Through detailed calculation, we determine each player's results based on their strategic decisions, which provides a basis for analyzing deviations and optimal strategies.

Deviation Analysis

Deviation analysis checks if any player can improve their payoff by changing their strategy, even slightly.
By diving into each player's current strategy, we identify potential deviations:

• Consider the chain-store: If the chain-store deviates from fighting to acquiescing earlier than specified, will it gain a higher payoff? This switch may affect challengers' decisions and subsequent payoffs.
• For challengers 1 to 90: If any challenger deviates by entering instead of staying out (or vice versa), would it lead to a better outcome?

Such analysis is essential for verifying whether the strategy pair forms a subgame perfect equilibrium. By confirming that no player can improve their outcome by deviating, we validate the optimality and stability of the strategies chosen.
This step involves meticulous evaluation of each possible alternative strategy and its impact on player payoffs.