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Puzzled: Lowest Number Wins

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Each of these puzzles involves a symmetric game. You will be asked about your best strategy, but what does "best strategy" mean? Here, we want a strategy that is a "Nash equilibrium" for all players; that is, one with the property that if it is followed by all other players, you can do no better than follow it yourself. Often, such a strategy requires that players do some randomization; for example, in the familiar game "Rock, Paper, Scissors," the Nash equilibrium strategy requires each player to choose rock, paper, or scissors with equal probability. As in "Rock, Paper, Scissors," plays in the games here are done simultaneously, with no collaboration allowed, so every man/woman for him/herself. For solutions and sources, see next month's column.


Joseph Skudlarek

Re: Footnote for "Coin Flipping" -- I setup and solved the recurrence relations for each entry in the 32x32 strategy matrix for the expected success for B and selected the maxi-min using Octave, a MATLAB clone; and then simulated to confirm. But it turns out this problem is the "Penney Ante" game described in "Concrete Mathematics" by Graham, Knuth, and Patashnik, Addison-Wesley (1989), p. 394, which you may want to look at for ideas and alternatives.

Joseph Skudlarek

My comment Re: Footnote for "Coin Flipping" has a typo: I solved for
Pr[A wins | row=choice(A), col=choice(B)], and took the maxi-min of
that; since B chooses first, she picks the min benefit to A for every
row=choice(A); then A chooses the max of all of those payoffs.

Other tools besides Octave that were helpful were C++ for simulation,
and Mathematica and SageMath for analytic (exact rational) solutions.

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