This game is called the Travelers' Dilemma. Two travelers,...

This game is called the Travelers' Dilemma. Two travelers, returning home from a tropical island where they both purchased an identical antique, discover that the airline has smashed these. The airline manager asks each traveler to write down the compensation desired, any whole number between $$\$ 2$$ and $$\$ 10$$ (the maximum the manager is allowed to give each traveler). If both write the same number, they each receive that amount. But if, say, $n_1>n_2$, then the manager gives each the smaller of the two (i.e., $n_2$ ) with an adjustment: believing that traveler 1 is lying and traveler 2 is being truthful, the manager punishes traveler 1 by giving $n_2-2$ as compensation, while rewarding traveler 2 for his supposed honesty by paying him $n_2+2$. (a) Draw the normal-form game when traveler 1 and 2 each write between $$\$ 2$$ and $$\$ 10$$. Why is $(10,10)$ not a NE? (b) What is the pure strategy Nash equilibria of this game?

This game is called the Travelers' Dilemma. Two travelers, returning home from a tropical island where they both purchased an identical antique, discover that the airline has smashed these. The airline manager asks each traveler to write down the compensation desired, any whole number between $$\$ 2$$ and $$\$ 10$$ (the maximum the manager is allowed to give each traveler). If both write the same number, they each receive that amount. But if, say, $n_1>n_2$, then the manager gives each the smaller of the two (i.e., $n_2$ ) with an adjustment: believing that traveler 1 is lying and traveler 2 is being truthful, the manager punishes traveler 1 by giving $n_2-2$ as compensation, while rewarding traveler 2 for his supposed honesty by paying him $n_2+2$.
(a) Draw the normal-form game when traveler 1 and 2 each write between $$\$ 2$$ and $$\$ 10$$. Why is $(10,10)$ not a NE?
(b) What is the pure strategy Nash equilibria of this game?

Key Concepts

Iterated Elimination of Dominated Strategies

Iterated elimination of dominated strategies is a method used to simplify games by progressively removing strategies that are inferior. This iterative process often leads to the identification of equilibrium outcomes, especially in games where the initial strategy sets are large or complex, such as in the Travelers' Dilemma.

Dominated Strategies

A dominated strategy is one that is strictly worse than another strategy for a player, regardless of what the opponents do. Recognizing and eliminating dominated strategies helps simplify the strategic analysis by narrowing down the choices to only potentially optimal strategies.

Nash Equilibrium

A Nash equilibrium is a set of strategies, one for each player, such that no player can benefit by unilaterally deviating from their own strategy given the strategies of the others. It provides a prediction of stable outcomes in strategic scenarios where each player is making the best decision they can, considering the actions of their opponents.

Normal-Form Game

A normal-form game is a way to represent a strategic situation by listing the players, their available strategies, and the payoffs associated with every combination of strategies. This representation is crucial in game theory because it allows for the systematic analysis of players' incentives and strategic choices in one-shot interactions.

Please give Ace some feedback