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.

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.