Two generals have 5 units each to deploy. Each person decides how many units to send to battle. The general who sends more troops will win, but it’s a draw if they both send the same number.

Each also has the option of “passing” which averts war and ends in a draw.

What’s the best way to play this game?

William Spaniel shows how to solve this game in his video series Game Theory 101. (video after the jump)

This is a nice explanation of the concept of using best responses to find Nash equilibria.

Video: Safety in numbers and best responses

If you liked this video, note that William Spaniel has written a great introduction to game theory which I have previously reviewed. Check it out.

If you’re never seen the show, here is how it works. Each of two contestants independently chooses to split or steal the final prize. If both choose split, then the prize is divided evenly. If one chooses split and the other steal, the person who steals gets the entire prize. If both choose steal, however, then both walk away with nothing.

Here’s the normal form representation of the game:

How should you play this game?

One contestant had an amazingly brilliant strategy that I will discuss below.

The wrong way to play the game

Contestants are allowed to discuss strategy before picking split or steal.

Both realize that split gives a fair 50 percent share to each side, but each also sees the advantage of back-stabbing and stealing the prize.

The discussion usually involves the following strategy. Each person tries to convince the other person to split, and they promise to do the same.

I discussed an example of this in a previous post: strategy in Golden Balls.

In that episode, both were promising they would split the prize, but then one person decided at the last minute to steal all the money. She said she was not proud of the decision, but she herself did not want to be cheated.

So trying to split the money in a conventional way doesn’t work. Is there a better strategy?

Why it’s bad to promise you will split

First, I want to explain why the strategy of splitting does not work. When you promise the other person you will split the prize, you are trying to change the game.

You are telling them that instead of looking at the original payoffs, they should only consider the game under the assumption that you are going to split the prize. So you are telling them to consider the following game:

Do you see what’s wrong with your strategy? If you promise them that you will split the prize, they are faced with a very tempting option to steal. If they split, they will only get 50 percent. But if they steal, they will get the entire 100 percent.

And therein lies the problem: if you promise you’ll SPLIT the prize, then you pretty much are telling them not to worry about the mutual steal option. This makes it a very good idea for them to STEAL the prize.

Clearly it’s a bad idea to promise that you’ll split the prize. Is there another way out?

How to beat the Prisoner’s Dilemma

There’s a remarkably devious way to get cooperation: you must tell them that you will STEAL the prize!

How does this strategy play out? You should watch the following clip to see the negotiation:

Brilliant strategy in Golden Balls

The action proceeds as follows. One contestant, Nick, immediately announces

I want you to trust me. 100 percent I am going to pick the steal ball. I want you to choose split, and I promise you that I will split the money with you [after the show].

The other contestant is completely stunned by this strategy, and the audience finds it amusing too.

The next 2 minutes is a funny exchange between the two. Nick keeps explaining why he is going to steal, and the other is dumbfounded by this terrorist-like action. He wonders, why can’t they both split?

The host reminds them the plan is risky, as there is no legal requirement for the money to be split after the show is over.

Nick is called an idiot and the other contestant just can’t believe he expects cooperation. Nick has taken control of the game, and he has not acted nice. Why should the other person cooperate?

Nick promises over and over that he is an honest person and that he will definitely split the money after the show.

At the 5:21 mark in the video, they reveal their choices. It turns out that both of them choose to SPLIT after all! Thus both end up with a 50 percent share of the money.

Here’s why Nick’s strategy was so brilliant. Nick was credibly explaining that he was going to steal. This changed the game into the following payoffs:

On the one hand, the other contestant could steal and destroy the prize money. On the other, he could split and hope that Nick kept to his word. In other words, Nick has transformed the game so that the weakly dominant best response is to split!

The other contestant is happy at the outcome, but he shouts “Why did you put me through that?” as he really had to struggle over his decision, only to learn Nick would cooperate after all.

I think Nick has shown a brilliant way to beat the Prisoner’s Dilemma in a one-shot game. His statement that he would steal is a credible threat, and it changed the game so the other contestant found split to be the appealing option.

This might not work in a repeated game, as Nick would have burned his reputation by not keeping to his word.

But in a one-shot game, it was a smart way to assure that both go away with a split of the prize.

The game works as follows. Alice will secretly write an even integer on a piece of paper, and Bob will secretly write an odd integer. Both are limited to writing numbers less than or equal to 1,000.

They will then simultaneously flip over their papers to reveal their numbers. The person who writes the lower number wins the game and is paid that number of dollars.

The game costs $10 to play and will only be played once. Should Alice and Bob give it a try?

What is the (subgame perfect) Nash equilibrium?

(credit: this game is from Tanya Khovanova’s Math Blog)

It could be a great game

If Alice and Bob could cooperate, the game would be extremely profitable.

If both wrote their maximum values–Alice 1,000 and Bob 999–then Bob would win $999 each time. Bob could then split the winnings with Alice. Both would profit $489.5 after each accounts for the $10 cost to play the game.

But amongst strangers such cooperation cannot be assumed. After Bob is paid, he could just as easily walk away from the table with $989 of profit and thank Alice for being a sucker.

So let us consider the case that neither person can trust the other.

Analyzing the incentives

Let’s say that Alice and Bob informally agree to write their maximum values, but each suspects the other will backstab and walk away with profits.

What is Alice’s best response to Bob writing the number 999?

If Alice writes 1,000, then she loses the game, and the best she can do is get $489.5, so long as Bob honors the deal.

Alice thinks: “What if I undercut Bob by writing 998 and try to win for myself? If Bob keeps to his agreement of writing 999, then I would win get paid $998 for undercutting, and I walk away with a lot more profit than our split of $489.5.”

Alice is not the only one who might worry about the arrangement. Bob could fear Alice might undercut him to 998, so he in turn will pre-empt and consider writing down the number 997.

You can probably deduce the rest of the story by extending the logic through backwards induction. If Alice fears Bob will write 997, then she is best to undercut once more to 996. But Bob could reason this far too, and he’s going to consider writing 995. As each person reasons this process further, they mutually undercut each other and the original agreement erodes.

So we can deduce each person reasons that it’s best to write a smaller and smaller number.

How low will the bidding go??

Finding the equilibrium

We can safely deduce Alice will eliminate large numbers, so she will never write a number larger than 10. Similarly, Bob will never write a number larger than 11.

But what happens then? If Alice writes 10, and Bob writes 11, then Alice ends up with $10 which just covers her cost to play the game. Her net profit is zero. Bob, on the other hand, loses his entire $10 entrance fee.

Bob, therefore, again has a reason to undercut Alice. He can think, “I can either write 11 and lose the game, costing me $10. Or I can instead write 9, which lets me win the game at a net loss of just $1. I know it’s a loss, but I’d rather accept a $1 loss than a $10 loss.”

If Bob is going to write 9, however, then what will Alice choose to do? She would naturally be better off writing the number 8. In that case she wins the game and nets a $2 loss, which is better than the $10 loss if she wrote 10 and let Bob win the game.

By continuing this reasoning, Alice and Bob are further tempted to undercut each other in an attempt to to minimize their losses.

Mercifully there is a limit to the madness. Once Bob gets down to writing 1, and Alice writes 0, the game ends with both parties winning nothing. [Edit 4-10: As Chris points out in the comments, the game probably ends when Alice writes 2 and Bob writes 1. There is no real reason for Alice to win the game with no profit.] At that point, Bob has no incentive to write a negative number (interpreted as him paying more to the casino).

The game theory equilibrium, therefore, is for Alice to write 2, Bob to write 1, and both of them end up winning nothing and each losing $10 for the priviledge of playing the game. This is not a pleasant outcome, but of course, that was the casino’s game all along.

Recently India caused a stir in global markets when it banned its cotton exports. The ban was a result of the unstable cotton global market, which has lead China to stockpile its cotton supply to insulate against price spirals.

The cotton market has been hit with surprises in the past year. The fear has lead to many countries pursuing a cautious strategy. The end result is worse for everyone. As explained in the Economist: “Behaviour that may be rational for individual actors can cause chaos if everyone copies it.”

Inspired by the situation, I came up with a simple game to illustrate how markets can fall apart in the face of fear.

The supply game

Suppose India and America each have strong and equal cotton supplies. Each country can choose to export some proportion of its cotton supply, represented as numbers x and y from the interval [0, 1].

Let’s say the global cotton market depends on India and America exporting sufficient cotton. Here are some contingencies and payoffs to the game.

–If the two countries supply enough cotton (say x + y > 0.5), then the global market is stable and both countries benefit equally for a payoff of +1.

–If the two countries do NOT supply enough cotton (say x + y <= 0.5), then the global market becomes unstable. A country that banned its exports gets the benefit of having a stockpile of domestic supplies in the unstable market (+2). A country that chose to export is hurt by the unstable market (-1).

How will this game play out?

Representing the payoffs graphically

There is a nice way to visually see the payoffs to the game. Here India’s choice to export is a number along the x-axis, and America’s choice on the y-axis. The payoffs are written as an ordered pair {India’s payoff, America’s payoff}.

The crucial line is the point at which x + y = 0.5. Anything closer to the origin is a point of market instability, and anything up and to its right is a point of market stability.

The dominant strategy

Here is how India might think about the game. We want to figure out its best response to any of America’s choices of exporting cotton. There are two cases to consider:

1. If America exports more than 0.5, the global market will be stable no matter what India does. India can choose anything from 0 to 1.
2. If America exports between 0 and 0.5, India can either supply a sufficient amount for market stability (+1), it can supply an insufficient amount (-1), or it can ban its exports out of fear and stockpile (+2). Remember choices are made simultaneously so India does not know of America’s choice beforehand.

The strategic thinking leads to a simple conclusion: with the chance of market instability, India is weakly better off banning its exports to protect its domestic supply.

America in turn will also feel the squeeze and be tempted to protect its supply, leading to both countries withholding cotton supplies and sending the market into a further price spiral.

The driving factor in this game is the fear of how markets will play out. When markets function smoothly, then countries are much happier to export and gain a higher profit (say +3) than they could get by only selling domestically.

Remedies to the game

In this game, the market for cotton completely fails. In actual markets, countries that withdraw their exports are seen with distrust and penalized in future interaction. India may have saved itself with cotton, but it did harm to its future trade profits.

As in the stag-hunt game, it takes coordination and solid communication to make sure everyone cooperates for mutual gain, rather than letting fear create strategically destructive decisions that cause market chaos.

The video is short and offers great practical advice for buying a car in America. Give it a watch:

Video: how to buy a car using game theory

And to pre-empt someone saying the video “is not really about game theory,” I respond that the following game theory concepts are used in the car-buying strategy:

–changing the game: you don’t go to the dealership, you call them

–auction theory: you are turning the negotiation process into an auction amongst dealerships

–credible threats: you walk out if they mislead you

Game theory can be a wonderful way of thinking about problems.