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Part I · The rules of the game

Being unpredictable

Matching Pennies has no equilibrium as long as each player picks one thing. The fix is to pick at random, with exactly the right odds. Finding those odds takes one line of algebra, and the answer explains penalty kicks, poker, and why a fair coin is the safest opponent in the world.

The rules of Matching Pennies: you and Robo each show a coin, Heads or Tails. If the coins match, Robo wins a point from you. If they are different, you win a point from Robo. Every box adds up to zero, so this is a zero-sum game: whatever you win, Robo loses.

Chapter 3 showed there is no settled box. If you always play Heads, Robo plays Heads and wins every time. So the first rule of this game is: do not be predictable. Test it. Play ten rounds against a Robo that flips a fair coin, then switch to a Robo that has a favorite, and see if you can take advantage.

Matching Pennies

+1 to you if different, +1 to Robo if the same.
Pick a side.
Rounds0
Your score0
Robo played Heads0

A fair coin has no pattern to find. A Robo with a favorite does.

Expected value

Suppose you pick Heads some fraction p of the time and Robo picks Heads some fraction q of the time. Both of you are now playing a mixed strategy: not one choice, but odds. What score should you expect per round, on average, over many rounds? That number is called the expected value. Move the sliders and watch it.

The odds of the odds

Probability is area: width times height.
Different: you winSame: Robo wins

Try this: set yourself to exactly 50%, then drag Robo's slider anywhere. What happens to your expected value? Then flip 100 coins at 50% for a star.

At exactly 50%, your expected value is 0 no matter what Robo does. Robo cannot outsmart a coin flip. And 50% is Robo's safe plan too. Half-and-half against half-and-half is the equilibrium of Matching Pennies, hiding outside the grid: neither player can do better by changing their odds. Nash's theorem promised it would be there.

Solving for the odds

Fifty-fifty is the answer for Matching Pennies because the game is perfectly symmetric. Most games are not. Here is a real one. You are taking a penalty kick; Robo is the goalkeeper. You kick Left or Right; Robo dives Left or Right. The numbers in the grid are the chance the ball goes in. You are a stronger kicker to the right, and a diving goalkeeper does not stop everything.

How often should you kick Left? Here is the trick, and it is the whole chapter: choose your odds so that Robo does not care which way it dives. If diving Left and diving Right give Robo the same result, Robo has no way to exploit you. Slide p until the two lines on the chart cross.

The penalty kick

Numbers are the chance of a goal, in percent.
Robo dives with its own best odds.
Math corner

Expected value. Multiply each outcome by its probability and add. In Matching Pennies your score per round is +1 with probability (different) and −1 with probability (same), so

EV = (+1) × P(different) + (−1) × P(same)

Probability is multiplication. If you pick Heads 1/2 of the time and Robo picks Heads 3/4 of the time, you both pick Heads 1/2 × 3/4 = 3/8 of the time. The square shows this as area. In letters, P(same) = pq + (1 − p)(1 − q).

The indifference trick, with the classic penalty numbers. If you kick Left with probability p, then

Robo dives Left: goal chance = 50p + 80(1 − p) = 80 − 30p
Robo dives Right: goal chance = 90p + 60(1 − p) = 60 + 30p

Set them equal: 80 − 30p = 60 + 30p, so 20 = 60p, so p = 1/3. Kick Left one time in three, and you score 70% no matter what Robo does. Doing the same algebra for Robo gives q = 1/2. The 70% is called the value of the game.

Von Neumann's theorem (1928). Every two-player zero-sum game has a value, and each player has a mixed strategy that guarantees it. This was the first big theorem of game theory, twenty-two years before Nash.

Big idea

When no single choice is safe, mix. Pick your odds so the other player gains nothing by favoring either of their options. Those odds are an equilibrium, they guarantee you the value of the game, and no amount of cleverness on the other side can take it away.

Try it on paper

1. A game pays you +2 with probability 1/3 and −1 with probability 2/3. What is your expected value per round?

Answer

(+2)(1/3) + (−1)(2/3) = 2/3 − 2/3 = 0. Fair, in the long run.

2. In Rock, Paper, Scissors, what mix makes Robo indifferent among all three of its choices?

Answer

1/3 each. Against that mix, Rock, Paper, and Scissors each win 1/3, lose 1/3, tie 1/3, so Robo has no favorite worth having.

3. Robo plays Heads 60% of the time in Matching Pennies. What should you do, and what is your expected value?

Answer

You want to be different, so play Tails every time. You win 60% and lose 40%: EV = 0.6 − 0.4 = +0.2 per round.

Challenge

Change Rock, Paper, Scissors so that winning with Rock counts double: +2 for Rock over Scissors (and −2 for the loser), every other win still +1. Find the mix that makes Robo indifferent. It is not 1/3 each.

Answer

Call your odds r, p, s. Robo's score playing Rock is p − 2s; playing Paper is s − r; playing Scissors is 2r − p. The game is symmetric so all three must equal 0: p = 2s, s = r, p = 2r. With r + p + s = 1 that gives r = 1/4, p = 1/2, s = 1/4. You play Paper half the time. Making Rock more valuable makes you play Paper more, because Paper is what beats Rock. Game theory is full of that kind of surprise.

True story

In 2003 the economist Ignacio Palacios-Huerta studied 1,417 penalty kicks from professional soccer leagues in Europe. Kickers and goalkeepers, who had never heard of von Neumann, mixed left and right almost exactly as the theory predicts: each kicker's scoring rate was the same whether he went left or right, which is what indifference looks like from the outside. They were also genuinely unpredictable from kick to kick, better than most people are when asked to "act random". Professional poker players do the same thing when they bluff a carefully chosen fraction of the time.

Stars in this chapter

Earn them by doing the clever thing, not by clicking around.