User:Cretog8/Scratchpad2
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The merged strategy (game theory) article has gone to its rightful place.
But here's some other thoughts on it.
Contents |
[edit] A disputed meaning
This seems a pretty good rough outline of trouble with mixed strategies, EXCEPT its a bit more about mixed-strategy Nash equilibrium than mixed strategies per se. Not sure the best way to separate that, or if it's entirely worth separating.
Need to attribute purification to Harsanyi, not Rubinstein
A third explanation for mixed strategies is that they exist only in the mind of other players. If a player playing Rock-paper-scissors believes the other player will play randomly, the game can still be studied in terms of mixed strategies. Aumann and Brandenburger describe Nash equilibrium in these terms as an equilibrium in beliefs, rather than an equilibrium in strategies.
Aumann, Robert & Brandenburger, Adam (1995), “Epistemic Conditions for Nash Equilibrium”, Econometrica 63: 1161-1180
[edit] stuff for information set
[edit] Example
At the right are two versions of the battle of the sexes (game theory) game, shown in extensive form.
The first game is simply sequential-when player 2 has the chance to move, they are aware of whether player 1 has chosen O(pera) or F(ootball).
The second game is also sequential, but the dotted line shows player 2's information set. This is the common way to show that when player 2 moves, they are not aware of what player 1 did.
This difference also leads to different predictions for the two games. In the first game, player 1 has the upper hand. They know that they can choose O(pera) safely because once player 2 knows that player 1 has chosen opera, player 2 would rather go along for 2 than choose f(ootball) and get 0. Formally, that's applying subgame perfection to solve the game.
In the second game, player 2 can't observe what player 1 did, so it might as well be a simultaneous game. So subgame perfection doesn't get us anything that Nash equilibrium can't get us, and we have the standard 3 possible equilibria:
- Both choose opera;
- both choose football;
- or both use a mixed strategy, with player 1 choosing O(pera) 3/5 of the time, and player 2 choosing f(ootball) 2/5 of the time.
[edit] stuff for monty hall
The second table in Why the probability is not 1/2 was un-intuitive for me
| Player picks Door 1 | ||||||
|---|---|---|---|---|---|---|
| Car behind Door 1 | Car behind Door 2 | Car behind Door 3 | ||||
| Host opens: | Door 2 | Door 3 | > Door 3 < | Door 3 | Door 2 | > Door 2 < |
| Host reveals: | Goat | Goat | > Goat < | Goat | Goat | > Goat < |
| Switching: | loses | loses | > wins < | wins | wins | > wins < |
