Question: The situation can be modeled as the game Peeking Pennies

The situation can be modeled as the game Peeking Pennies. This game is the same as Matching Pennies, except that Eve receives a signal after Adam’s choice, which says ‘‘Adam chose heads’’ or ‘‘Adam chose tails.’’ It is common knowedge that the message is correct with probability h when Adam chooses heads and with probability t when he chooses tails. If h > t and h þ t > 1, show that there is a Nash equilibrium in which Eve always chooses tails when she hears the message ‘‘Adam chose tails,’’ but the players otherwise mix their strategies. Confirm that Adam’s probability of winning in this equilibrium is less than his probability 1 2 of winning in regular Matching Pennies. a. Why is Peeking Pennies relevant to the Ellsberg Paradox? b. What happens when we erode Eve’s predictive power by allowing h and t to approach 1 2? c. What happens if we try to instantiate the Newcomb’s Paradox of the philosophical literature by taking h ¼ t ¼ 1?

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Introduction

The game theory model "Peeking Pennies" presents a strategic interaction between two players, Adam and Eve, with an added layer of information asymmetry compared to the classic "Matching Pennies" game. In this model, Eve receives a signal after Adam makes his choice, which indicates Adam's choice with certain probabilities. The question of interest involves determining whether a Nash equilibrium exists under specific conditions, analyzing Adam's chances of winning in this equilibrium, and relating the game to well-known paradoxes in decision theory, such as the Ellsberg Paradox and Newcomb's Paradox. This essay aims to demonstrate the Nash equilibrium in "Peeking Pennies," explore its implications for Adam’s probability of winning, and discuss the relevance of the model to the Ellsberg and Newcomb paradoxes. The analysis will be structured to provide a theoretical understanding of the model, supported by mathematical arguments and examples where applicable.

Nash Equilibrium in Peeking Pennies

To understand the Nash equilibrium in the game of Peeking Pennies, we must first recognize how it extends the classical Matching Pennies game. In Matching Pennies, both players—Adam and Eve—choose either "heads" or "tails" simultaneously, and if their choices match, Adam wins; otherwise, Eve wins. This game has a mixed-strategy Nash equilibrium where both players choose heads or tails with equal probability (1/2), leading to a 50% chance of winning for both players.

Peeking Pennies, however, introduces an asymmetry by allowing Eve to receive a signal after Adam makes his choice. The signal informs Eve about Adam's choice with a probability hhh if Adam chose heads, and ttt if he chose tails. The signal, therefore, provides Eve with an advantage, albeit an imperfect one.