I suspect you Candidate Masters worked it out. If you were playing France, what would you do? An answer like this is acceptable: "I would play FS for a whole lot of moves. If Austria ever played AC, I would win. If he kept playing AD, the position would not change. After many moves, I would try to sneak in an FA. Unless Austria guessed almost psychically, he would not play his AC exactly at the moment I played FA. So I would win, either with one of my many FS's against an AC, or with my one FA against one of Austria's AD's." That is a good answer. Practically speaking, you can do extremely well that way. Now let's compose a proper, rigorous answer to the question. (This is a Master Class, after all.)

Players choose moves. They choose their moves based on the position on the board, of course. They also choose moves based on their views of other players' psychologies. But we do not need to speculate about psychologies for our analysis. In the world of game theory, players choose Strategies. Some strategies are moves, but some are not. A Mixed Strategy is a probability distribution of moves. For example, here is a mixed strategy for Austria: Play AC with probability .5, otherwise play AD. That strategy can be implemented by flipping a coin and playing AC if it comes up heads, AD if it comes up tails. Let's write Mixed Strategies like this: [S1, .7; S2, .3] We mean that the move S1 is played with probability .7, and the move S2 is played otherwise, with probability .3. (Note for cognescenti: we are using an oversimplified notion of a Mixed Strategy here.)

We can specify an extremely good Mixed Strategy for France: [FS, .9; FA, .1] Then Austria can do no better than to play AD on the very first move (that is, there is no Mixed Strategy that does better against the given Mixed Strategy for France). And this gives him only a .1 chance of reaching a draw. Of course, there is an even better mixed strategy for France: [FS, .99; FA, .01]. And a better one than that. Is there a best strategy for France? Simply playing FS with probability 1 is not a best strategy for France. For in that case, there is a sound reply by Austria: he plays AD with probability 1. And then the game drags on ad infinitum. There is no best strategy for France. But, France can make her probability of winning as high as she likes, up to but not including 1. This is the sort of situation that I call a "virtual force." One side can choose from a menu of strategies, and if patient can bring the chance of winning as close to 1 as desired.

Thus endeth the lesson. The particular endgame position I chose seems to be fairly common. But of course, the main ideas generalize to other endgame positions. I am always interested to see examples, so let me know if you run across one.

Two possible topics for another class:

Mixing elements of chance into an endgame position. Do you have a .5 chance of winning? How can we measure precisely?

Second-order virtual forces (a discovery of Dan Shoham's, developed by me). Exotic topic, not terribly practical but theoretically beautiful. In theory, but not on the Diplomacy board as it is currently known, there are also higher-order virtual forces.

Class dismissed.

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