Question

Player 1 writes an integer between 1 and 15 (including 1 and 15) on a slip of paper. Without showing this slip of paper to Player 2, Player 1 tells Player 2 what they have written. Player 1 may lie or tell the truth. Player 2 must then guess whether or not Player 1 has told the truth. If caught in a lie, Player 1 must pay $10 to Player 2; if falsely accused of lying, Player 1 collects $5 from Player 2. If Player 1 tells the truth and Player 2 guesses that Player 1 has told the truth, then Player 1 must pay $1 to Player 2. If Player 1 lies and Player 2 does not guess that Player 1 has lied, then Player 1 wins $5 from Player 2. Determine the von Neumann value of the game and optimal strategies for both players.

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Adonis

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74 Answers

The calculation shows that Player 1's optimal strategy is to always tell the truth (100% of the time), as indicated by the strategy array
[
1
,
0
]
[1,0], where the first element corresponds to telling the truth and the second to lying. This result suggests that, in the optimal mixed strategy, Player 1 does not benefit from lying within the structure of this specific game.
The von Neumann value of the game, from Player 1's perspective, is $1. This value represents the expected amount that Player 1 would have to pay to Player 2 per game, on average, when both players use their optimal strategies.
Given this, the optimal strategy for Player 2 would be to always guess that Player 1 is telling the truth since Player 1's optimal strategy is to never lie. Under these optimal strategies:
If Player 1 tells the truth (which they always do), and Player 2 guesses that Player 1 has told the truth, then Player 1 must pay $1 to Player 2.
These strategies and outcomes ensure that both players cannot improve their situation by unilaterally changing their strategies, hence achieving equilibrium.

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