Dixit’s Dicey Digits

The secret of strategic thinking revealed.

Posted Jul 16, 2019

Gearstd/Shutterstock
Source: Gearstd/Shutterstock

You have to surprise opponents, keep them guessing. Doing the same thing over and over again without variation will not work. —A. Robben, soccer player

The whole art of war consists of guessing at what is on the other side of the hill. —The Duke of Wellington, trying to say something deep about strategy. This answer is of course "Old Age."

In The Art of Strategy, Dixit & Nalebuff (DN) bring game theory to life, offering to help us better understand the uncertainties of human interaction and giving us some tools to deal with them. Their opening game is to have us pick a number between 1 and 100. DN also picked a number. If we guess their number, they will pay us a $100, otherwise nothing, but if we get it wrong, they will tell us if our guess was too high or too low. We can then guess again, but this time, they will only pay us $80. The next time, they’ll pay $60, then $40, and finally $20. If after five rounds we still haven’t guessed their number, well, then we’re out of luck.

Let’s suppose DN picked their number randomly, what is our best approach? Given our opportunity to narrow down the range of possibility over successive rounds, we should split the range in the middle with each guess. Our first guess of 50 will be correct with a probability of .01, but if it isn’t, we have improved our chances for the next round as much as possible. Then, we have a probability of winning of .02, as opposed to, say, .0103, if we had guessed 98 and learned that this guess was too high; and this guess is more likely too high than too low. 

J. Krueger, N. Thakral
The expected value of the Dixit game
Source: J. Krueger, N. Thakral

The worst outcome is not to win anything. But what if DN offered us a lottery (they do not in the book)? How much would we pay so we could play? What cost would we accept to receive a possible but uncertain benefit? If we are not risk-averse, we’d pay any price lower than the game’s expected value, EV. The EV for the first round is $1, that is, .01 x $100. For the second round it is $1.584, or .99 x .02 x $ 80; for the third round it is $2.32848, or .99 x .98 x .04 x $60; for the fourth it is $2.9804544, or .99 x .98 x .96 x .08 x $40; and for the fifth round it is $2.74201805, or .99 x .98 x .96 x .92 x .16 x $25. The EV of the entire game is the sum of these roundwise EVs, namely $10.6349524. See also the calculation above or behind this link.

The one-round equivalent of this five-round game is winning $37.9521179 with a probability of .28022026. We might be willing to pay $10, knowing that the game’s EV is 63c higher. Of course, most people prefer sure money over probable money. If we could play this game only once, most of us would rather keep our $10, but if the game were played many times, we’d know that we’d win eventually. The longer the run, the more certain the win. The casino would be, in fact, us, and we’d pay more to be the casino, the longer we’d be allowed to exploit the sucker. For the single-round game, however, we’d pay less than the EV in order play, but how much we'd pay is not a matter of objective calculation but a reflection of the degree of our risk aversion. Our willingness to pay would reveal our risk attitude.

Might this attitude differ depending on whether we play DN’s five-round game or the compacted one-round game? The EV is the same. Chances are we’d be willing to pay more for the five-round game. Why? The five-round game is more appealing in two ways: there are more opportunities to win, and 4 of the possible wins are greater than the $37.95 pot in the single-round game. It’s just that the individual probabilities of winning are much smaller. But we tend to overestimate very small probabilities as well as disjunctive probabilities, that is, the probability of winning once in up to five attempts. This is why many of us buy lottery tickets. We focus on the large pot and underestimate the probability that we will always lose (= we overestimate the probability that we will win at least once).

For DN, the lottery game is just a warm-up. It has no strategic element. Judgment is just a matter of working out our preferences and getting a grip on the probabilities. But DN are interested in The Art of Strategy. In the interpersonal version of the game, they pick a number after considering how we, their opponents, might go about guessing it. It’s a game of hide-and-seek. Their objective is to hide their number where we are least likely to look. Such games of strategy open up rabbit holes because the winner is the one who manages to look deeper into the hole than the opponent.

DN might begin by assuming that we use the method of range-splitting as we move through the rounds, generating estimates of, say, 50, 75, 87, 93, and 96. If they assume we will use this strategy, they will not pick these particular numbers. They will thereby attempt to ensure that if we get lucky at all, it can only be during the last round, for the measly payoff of $20. If we, in turn, realize that DN will be attracted to this strategy, we can respond by jiggling our range-splitting strategy and offer numbers such as 49, 76, 85, and so forth. Small jiggles will not reduce the EV by much, that is, they will keep it close to $10.65. So what is it worth to us? Will we offer to pay more (or less) for a game of strategy than for a game of risk?

A rule of thumb is that those who propose a zero-sum game think they can outsmart those who take it; and those who accept the game think they can outsmart those who offer it. In zero-sum games such as this one, no value is created by the transaction. One party’s gain is the other’s loss. In this world, overconfidence keeps things moving by re-distributing wealth. If overconfidence leaves a footprint here, it will be seen in our willingness to pay more for a strategic game than for a mere game of risk.

What about the rabbit hole? Game theorists often recommend strategies to be randomized when there is no good reason – and hubris is not a good reason – to think one is cleverer than the other. Randomization at least offers some protection against exploitation. This would be good advice if there were only one round in the game. With multiple rounds, we end up recommending ‘the jiggle,’ but we have nothing to say about the size of this jiggle.

References

Dixit, A. K., & Nalebuff, B. J. (2008). The art of strategy: A game theorist’s guide to success in business and life. New York: Norton.