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%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.
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.