## Posted tagged ‘game theory’

### Worse is better

May 28, 2012

As I promised in my last post, I will give two more examples of games, where it is better to be the weaker/less capable player.

The first game is three person Russian roulette: Three persons roll a dice to decide who is player A, B and C. First player A tries to shoot one of his opponents, if player B is still alive he tried to shot one of his opponents, then it is player C’s turn if he is still alive, after that it is player A’s turn and so on, until only one person is left. We assume that all shots are either lethal or a miss and we require all player to do their best at each shoot: you are not allowed to miss on purpose. If all three contestants have 100% chance of hitting on each shot, player A will kill one of B and C, and the survivor will kill player A. If A chooses his victim randomly (to be correct: with uniform distribution) then player B and C each have 50% chance of winning, and player A will die. However, if player B has only 99% chance of succeeding at a shot – and his opponents knows that – then player A will shoot player C, to get a 1% chance of surviving. Then player B has a 99% chance of winning. Similarly, if player C has 99% chance of succeeding on each shot, and A and B has 100%, then player C will win with probability 99%. Finally, consider the case where player A has 99% chance of hitting and the two others have 100%. If player A misses his first shot, he will be in the same situation as if he was player C, so he will win with probability 99%. If he hits in his first shot, the other player will hit him. Thus player A has 1%$\cdot$99%=0.99% chance of surviving. So if you have 99% chance of hitting on each shot, both you opponents  have 100% chance and everyone know this, then you will survive with probability 66.33%.

This game has another “paradox”. If you are player A, you hope to miss your first shot. If we change the rules to allow the player to miss on purpose (and to avoid a stalemate, let’s stay that everyone dies if there are three misses in a row) the player with 99% chance of succeeding at each shot will survive with probability 99%, while each perfect shooter will only survive with a probability of 0.5%.

It might not be surprising that the weakest player can have an advantage in three person games: The two best compete against each other and then the weakest player can attack the survivor of the two strongest. I think the game from the last post was more surprising. Here two players play chicken, but one of the players, let’s call him player B, cannot turn the steering wheel. If player B could turn the steeling wheel, there would be two Nash equilibria, one where player A wins the most and one where player B wins the most. But when player B cannot swerve, the Nash equilibrium where A wins the most disappears, and we end up in the Nash equilibrium where B wins the most.

I find the next game even more surprising. This game also have two players, A and B with A being stronger that B, but now both A and B can “do the same to the environment” and player A can choose to use his strength against player B. The game is played by two pigs. They are trained separately to press a panel in one end of their sty to get food in a feeding bowl in the other end of the sty. We then put both pigs in the sty together. We assume that the dominant pig, A, can push B away from the feeding bowl, but he cannot hurt B.  If B presses the panel, A will be closer to the food bowl, and B is not strong enough to push him away, so B does not have any reason to push the panel. One the other hand, if A pushes the panel, B will eat some of the food, but A can push him away. If they get enough food for each press on the panel, there will be food left for A, so he will start running back and forth between the panel and the food bowl, while B will be standing close to the food bowl all the time. If they do not get too much food for each press on the panel, B will get more food and A.

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I remember that I have heard about the three person Russian roulette, but a cannot find any references now (added later: a reader pointed out that this game was mentioned here in the quiz show QI). The game with the two pigs is described an article by Baldwin and Meese (but it is older). They tried this experiment, but it was a box of length 2.8 m so the dominant pig got the most food. I do not know if there are experiments that show that a dominant animal would do the panel pressing if it gets less food than it opponent.

Baldwin, B. A. & Meese, G. B. 1979. Social behaviour in pigs studied by means of operant conditioning. Animal Behaviour, Vol .27 Part 3, pp. 947–957.

### How to get one dollar for only a few cents

May 24, 2012

Most of this post is a translation of my article in Famøs (a student journal at University of Copenhagen). If you have already read that article, you might want to jump to the puzzle.

The ”Dollar Auction game” is a very simple game: An auctioneer wants to sell one dollar to the highest bidder, but there is one unusual rule in this auction: Both the highest and the second highest bidder have to pay their bid, but only the highest bidder will get the dollar. All bids have to be in multiples of one cent. What would you do in this game?

Let’s see what happens if you play this game with a lot of people. It only cost 1 cent to give the first bid, and it could earn you 1 dollar, so probably someone will give that bid. But then 2 cents for a dollar is also a good deal, so someone else bets 2 cent. Then someone bets 3 cent and so on. Now, let’s say the Alice bet 98 cents and Bob has just bid 99 cents. If Alice stops here, Bob will get the dollar for 99 cents, so he earn 1 cent, but Alice will have to pay 98 cent. To avoid this, Alice bids 1 dollar and if Bob stop here, Alice get the dollar for one dollar, so she don’t lose anything. However, Bob don’t want to stop because he will then loose 99 cents, so instead he bids $1,01, hopping that Alice stop and that he will only loose one cent. So Alice and Bob will continue the bid for a while, until one of them give up. I have never tried this game, but there are claims that someone paid$200 for one dollar, or even $3000 for$100, so you shouldn’t try this at home,… but perhaps you should try it somewhere else as the auctioneer!

What should you do if someone started a Dollar Auction? One strategy would be not to bid at all, but that it too boring. Another strategy is to explain the problem to everyone and then bid 1 cent hoping that no one else bids… or at least, hope that the probability that no one else bids is less than 99%. A third strategy is to bid 99 cents before anyone else bids. This way, no one has a reason to overbid you. There are two problems with this strategy: Even in the best case, you can only earn 1 cent with this strategy and if some in the crowd really hates you, he can bid 1 dollar just to make you lose the 99 cents!

A more interesting strategy would be to bid 1 cent and promise that you would not let anyone else get the dollar for less than $1.02. If all others really believed you, they should not bid. But why should they believe you? If Alice bids 99 cent after you have bid 1 cent, it would probably be best for you to just break your promise. Alice would then earn one cent, and you are the only one who loses, so no one will be mad at you for breaking your promise. This leads us to a counterintuitive strategy. Make a deal with Bob: “If someone else gets the dollar for less than$1.02, I have to pay you $3”* After making this deal, you bet one cent. If someone, say Alice, bets 99 cents, it will be better for you to bet higher, that to give Bob the 3$. Alice know this, so she will not try to bet higher.

So the strategy is to promise to give away some money under certain conditions. Intuitively, you would think that this is a bad idea because you are restricting yourself.  However, in some games it is best to make a “voluntary but irreversible sacrifice of freedom of choice”** If you play chicken (a game where two drivers drive their car against each other, if you swerve you lose, and if none of you swerve you probably gets killed or injured so that also counts as losing) you are almost sure to win, if you, before the game starts, take of your steering wheel. Your opponent knows that you cannot swerve, so he will have to swerve. However, it is important to tell your opponent that you cannot swerve, otherwise it might end in disaster, as in the film Dr. Stangelove! [SK]

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Puzzle: How many “essentially different” games can you find, where it is best to be the weakest/less capable player?

I know that statement is a bit weak, but I didn’t want to make it too precise. From the above we can find one example: If you play chicken it will be an advantage to not be able to move your arms (and not be able to turn a steering wheel in any other way) as long as your opponent knows that you cannot swerve. I consider many games to be “essentially” the same as this game, although I am not able to define the class of games that I consider to be essentially the same. I have two other essentially different games where it is an advantage to be the weakest/less capable, and I will post them next week.

* Actually, this is not a good deal, because it will be possible to use it against you to blackmail you.  Furthermore, Bob should know that you would never get anyone else get the dollar for less that $1.02, so he would never earn the$3. A better deal would be the following (I hope!) “If someone else get the dollar for less than $1.02, or if I make any other agreement during this game, or pay or receive money during this game or as a consequence of this game I have to pay you$3. If you make any other deals during the game, you have to pay me \$10. You get 10 cents for accepting this deal.”

**This phrase is from the Nobel prize winner Thomas Schelling [TS]. Steven Pinker gives other examples of such games in [SP, p. 408-411].

[MK] Muringhan, J. Keith.  “A Very Extreme Case of the Dollar Auction.” Journal of Management Education 26, 56-69. 2002

[TS] T. C. Schelling, The strategy of conflict, Harvard University Press, 1980.

[SL] S. Kubrick, Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb [film], Columbia Pictures, 1964.

[SP] S. Pinker, How the mind works, Norton, 1997.