Game Development Reference

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his strategy to play paper 100% of the time. When one player can gain by

changing his own strategy, the configuration is not a Nash equilibrium.

Mixed equilibria are easy to work out in simple games like rock-paper-

scissors and matching pennies, where one player wins outright while the

other loses. But this is an unusual case. In most real strategy interactions,

different outcomes have different payoffs. For example, in a fighting game,

a block beats a jab while doing no damage, a jab beats a throw while doing

a little bit of damage, and a throw beats a block while doing lots of damage.

This is analogous to a version of rock-paper-scissors where you get $1 if you

win with paper or scissors, but you get $5 if you win with rock. The payoff

matrix looks like this:

A naïve strategy would be to simply play rock every game and hope for

the $5. The problem with this is that it is predictable. The opponent can

counter by playing nothing but paper, and you'll walk away with nothing.

To play well at this game, you need to play a mixed strategy that randomly

chooses among rock, paper, and scissors. But you can't just play them

evenly as in vanilla RPS, because your opponent will respond to you by

playing rock more often. So how often do you play each move to maximize

your earnings?

This is where the mathematical aspect of game theory comes into

play. Given a strategy interaction and a set of payoffs, game theorists can

calculate the precise proportions of a mixed strategy that creates a Nash

equilibrium. Game designers don't need to do this numerically, but un-

derstanding how the proportions relate is important, so I'll demonstrate it

with a real-life example.

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