[danteembermage]

Think about it this way: Suppose we are playing battleship on a board with one patrol boat and five squares in a line. There are only five possible places to put a patrol boat on this board. The three middle squares are equally likely to be chosen by a random arrangement, so we might as well choose the middle square for an opening move.

This is where this falls down. Since I know it's optimal for you to choose one of those three middle squares, I do not want any boat configuration that covers two of them, so I will always pick either the far left patrol boat or the far right patrol boat.

Except if I always pick the far left or right, why would you ever pick middle to start? You wouldn't. Finding an equilibrium to these games is hard, but I'd guess a 50/50 mixed strategy of placing a boat either on the extreme right or extreme left and a 50/50 guessing strategy of B1 or D1 would be a Nash Equilibrium of this game. And the boat arrangement would look nothing like what you'd expect from a random arrangement, in fact there'd be a giant black probability square right in the middle of the board.

[Triumvark]

I like your simplified model, and I know random strategies are sometimes helpful. My point was that random strategies are not always helpful.

I worry people don't believe this, so let me use a silly outlandish example to prove my point.

Here's a game: Suppose we have a countably infinite number of boxes, all numbered. You pick one to hide under, and I have to find you by looking under boxes which I choose, one at a time. I get to look at as many boxes as I like until I find you or get tired and go home.

Under box 1 is famed astronomer Neil deGrasse Tyson, who was spying on you with his (perhaps magical) telescope all morning. He will gladly tell me where you hid.

I now announce my strategy for this game, namely, to pick Box 1 and ask Tyson how to win.

Will you now respond with the original, "From the point of view of game theory, this is not an optimal strategy since it merely invites the opponent to avoid box 1. ..."?

The point of this silly example game is that randomized strategies aren't necessarily helpful for certain classes of games where one strategy gives you disproportionate information about the state of the game.

Guess my number games and battleship games risk being very similar to my extreme illustrative example, but I'll admit I don't know how much.

Bottom line: I just resent the insistence that good game theorists dogmatically REQUIRE randomness in all competitive situations. This is not necessarily the case.

[danteembermage]

In my experience mixed strategies are very important to pushing the boundaries of game theory but are rarely important in applying it (I come from business/economics so that biases the kinds of models that are useful). For example, when we think about how to optimally design an executive compensation contract the resulting equilibrium usually has no randomness at all or at least the actors are not playing a mixed strategy. The executive knows what "effort" level she will exert and the compensation committee knows what it will pay her (not in absolute dollar terms but in terms of an explicit contract). The only mixing example I can think of is that Soccer goalies appear to be mixing with which direction they dive and kickers kick http://www2.owen.vanderbilt.edu/mike.shor/courses/game-theor... although I think duopoly product competition is often viewed that way (not competing on price or production size but whether to launch a new product)

[feral]

Obviously, if there is a single move that wins you the game every time - being able to consult the oracle, in your example - then this pure strategy is dominant, and no mixed strategy is called for.

http://en.wikipedia.org/wiki/Pure_strategy#Pure_and_mixed_st... http://en.wikipedia.org/wiki/Strategic_dominance

It is obvious that if there is a dominant pure strategy, then that is the strategy to play, and there is no need for a mixed strategy.

Anyone with elementary game theory knows this, and, really, no 'good game theorist' is going to 'dogmatically' assert otherwise.