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« on: July 29, 2012, 07:55:05 pm »
"obvious play is obvious" - theorel
the big tl;dr: the word WIFOM is used too much, and many allegedly WIFOM situations are in fact quite concretely solvable.
Here's a modified WIFOM game. (It's kinda lame, cause nobody dies.) I'll call it false WIFOM. We sit in front of a table with a green glass and a white glass. I secretly write down a word "green" or "white" on a piece of paper to "poison" that glass. You choose a glass and "drink" from it. If you sip from the poison glass, you have to give me ten dollars. However, if you drink from the green glass, I have to give you five dollars (whether it was was poisoned or not).
At first glance, this seems similar to the classical WIFOM problem (same game without the green glass twist). The drinker's strategy should be "go for the green glass," since it has the bonus, except, wait, the poisoner knows that, so the poisoner will just poison the green glass, so you should really go for the white glass, except, wait...
However, I claim there is a subtle difference. To see this it's helpful to employ the language of "mixed strategy." In a "mixed strategy," both players make their choices based on some sort of randomness, like, say, a die roll. So the poisoner might flip a coin and poison if the coin turns up heads, say.
An "equilibrium solution" to a game (for the purpose of this post) is a mixed strategy for both players that neither player can gain by deviating from (unless the other also deviates). In other words, the equilibrium solution yields the maximum score that one player can make given that his opponent plays perfectly, and vice-versa.
It turns out that there's a definitive answer to the question: which mixed strategy should the players adopt going into this game? (At least if by that we mean "find an equilibrium solution.") The poisoner should poison the green glass with probability 3/4, and the drinker should pick the green glass with probability 1/2. Two proofs below, spoilered for people who don't like math.
Proof (with calculus): The second person's expected gain from the game (which is actually a loss) is a function of two variables: x, the probability that the poisoner poisons the green glass, and y, the probability that the drinker drinks from the green glass. More precisely,
P(x, y) = y(5 - 10x) - 10(1-y)(1-x)
The equilibrium solution is the "saddle point" of this function. In other words, it's the point on the graph where, if x is fixed, and y is allowed to vary, P(x, y) only goes down, but if y is fixed, and x is allowed to vary, P(x, y) only goes up. It's a straightforward calculation to see that x = 3/4 and y = 1/2.
Second "proof" (just algebra, but an intuitive argument that I find a little sketchy): The first player, intuitively, should pick a strategy that leaves the second player with no real choice, i.e. should put the second player in a classical WIFOM position. If he poisons the green glass with probability x, the expected payoff to the second player for picking that glass is 5 - 10x. The expected payoff for the second player picking the white glass is -10(1-x), and he wants these numbers to be equal.
What's the moral? Well, if you're the guy drinking the glass, there is no moral. The other player can pick an optimal strategy that makes you 50-50, and so you may as well have been playing literal WIFOM. You might conclude that this analysis has no bearing on the game of mafia, but I don't agree.
Here's why: let's say you're a fly on the wall in the modified WIFOM game. You're not going to drink a glass or poison a glass; you're just watching. You have a side bet with another fly on the wall. Your bet is: which glass did the poisoner put the poison in? What should you bet? Of course you should bet he put it in the green glass (since in the equilibrium solution, he did so 75% of the time). The second player still picks the green glass sometimes because of the shift of the risk-reward calculation, but it doesn't change the basic fact that the green glass is more likely to be poisoned.
In Mafia, your role is part glass-drinker, part fly-on-the-wall. Sometimes (e.g. LyLo) you really are in classic zero-information WIFOM - they've put you in a position that you have to make a random guess. But if there's a lot of game left, you learn information about other players from moves they've made. So when you see someone make a move that's actively bad for scum (i.e. it would be hurting his own team if he were scum), you should think "I've gained some information that he is less likely to be scum" in spite of the WIFOM. Likewise, when you see someone make a move that's actively good for scum, you should think "he's more likely to be scum," not "scum would never do that, it's too obvious, unless that's what scum WANTS me to think, unless . . ."
