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This topic in game theory can be solved using only algebra. I performed a complete solution analytically, then used a computer program to check my analytical results.

High Card Wins

The following game is a variation of a puzzle that I found in Michael Shackleford's Math Problems site. Problem 60 reads as follows

Computer helpful. Two players are each dealt a card face down. Each player may look at his own card. The highest card wins. Cards are valued as in poker with aces being low. The first player may either keep his card or switch with the second player. The second player may keep his card, whether it be his original card or one that the first player gave him after switching, or trade it with the next card on the deck, which is also face down. The loser pays the winner $1 and if both cards are equal then no money exchanges hands. For the sake of simplicity assume an infinite number of decks. Both players are infinitely logical. At what point should the first player switch? At what point should the second player switch if the first player doesn't switch? What is the expected gain of the first player?

The game that I want to consider is only slightly different. Rather than have all ties end in a draw, the second player wins all ties, with just one exception. If the first player elects to trade cards and they turn out to be equal, then player 2 must draw the third card in an attempt to break the tie. This slight rule change makes a profound difference in the solution, and that difference is the entire point of this topic.

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