Talk:Mixed strategy

Removed line

I have removed the following line from the previous entry:

A player would only use a mixed strategy when they are indifferent between several pure strategies, and when keeping the opponent guessing is desirable, that is, when the opponent can benefit from knowing the next move.

I'm worried this might be misleading. Supposing that players only play Nash equilibria, this statement is true. But an entry on mixed strategies should not assume that players only play Nash equilibria, one might play a mixed strategy which is not Nash. --Kzollman 21:14, Mar 17, 2005 (UTC)

I think that mixed strategies can be useful even if you don't look for/have the option of playing Nash Equilibria. I added a few sentences to the Significance section to note three situations where you might use a mixed strategy. I included a reference to back it up, but if anyone has more comments that would be great. The Strategies and Games book has examples for each but I don't have the time to type them out right now. ... I think you could argue that not playing a Nash equilibrium when one is available isn't rational, and as this is an article on game theory you could assume that all players act rationally. --Culix 03:50, 5 April 2007 (UTC)[reply]

Example optimal mixed/pure strategies

These tables represent the payoff matrices for player A along with the optimal strategies in column A_p representing the probability of each move, with the table captioning "strategy N" where N represents the minimum expected payoff for player A.

*mixed strategy -0.2*
	A_p	B₁	B₂	B₃
A₁	0.2	3	-1	2
A₂	0.8	-1	0	3
A₃	0.0	-4	-3	1

*pure strategy -4*
	A_p	B₁	B₂	B₃
A₁	0.0	-9	-4	8
A₂	1.0	-4	7	7
A₃	0.0	-5	0	1

*mixed strategy 0.937*
	A_p	B₁	B₂	B₃
A₁	0.0	6	0	-1
A₂	0.188	-8	5	-8
A₃	0.812	3	0	3

*mixed strategy 1.681*
	A_p	B₁	B₂	B₃
A₁	0.0	6	2	-2
A₂	0.437	9	-1	0
A₃	0.563	-4	8	3

If you have any questions please ask. --ANONYMOUS COWARD0xC0DE 04:13, 2 January 2007 (UTC)[reply]

I reverted because these because they add no explanatory power to the article. Why are there four matrices? What property to the four examples illustrate that is particular to each of the four? We are to assume thes payoffs given are for player A? What are the payoffs to player B? The Ap column isn't part of a payoff matrix, so what is it, the probabilty of using that move at the mixed Nash? What is the number in table footer? Why do you expect readers to supply their own answers to these questions? Without adequate explanation these examples are incomprehensible to the reader, you cannot expect readers to decipher totally opaque material. The onus is on the writer, not the reader, to provide the thinking. Pete.Hurd 16:41, 5 February 2007 (UTC)[reply]

Meaning

I have just added the pragraph about the meaning of mixed strategies. I believe it factually correct, but I know my English is not as good as it should be to contribute to this Wikipedia. Hence, all corrections are welcome.

On a more fundamental level, I am an Industrial organization guy, not a game theorist. Rubinstein and Aumann are proeminent figures of game theory, but some of their claims about its meaning can be controversial. Hence, this paragraph may be biased. All I can describe is the impact of this insider criticism upon the use of mixed strategies in economic research at large. Again, comments and corrections welcome.--Bokken | 木刀 09:45, 10 March 2007 (UTC)[reply]

Kudos for using references to back it up though. I think I found a copy of Aumann's paper here. If anyone wants to dispute that it's not the real thing, feel free. --Culix 03:42, 5 April 2007 (UTC)[reply]

another removed line

"Finally, mixed strategies are useful because an opponent is less likely to correctly guess your move, which can also help a player gain a higher-than-average payoff." Since both players are rational and everyone knows that, wouldn't the other player be able to correctly guess your strategy, in this case a mixed one?66.156.90.250 20:16, 30 May 2007 (UTC)[reply]

"Mixed strategies are important in game theory because they can allow players to reach a Nash Equilibrium when one would not normally exist. Also, a mixed strategy can sometimes allow a player to attain a higher expected payoff than choosing any available pure strategy. (Dutta:103-115)" Mixed strategies are important, but they exist "normally", an equilibrium is a bit more sophisticated concept than "sometimes allow a player to attain a higher expected payoff". Pete.Hurd 20:43, 30 May 2007 (UTC)[reply]

I agree, and I think this is a bit misleading, since a mixed strategy cannot give you a higher payoff against a set opponent's strategy profile than any of the pure strategies which compose that mixed strategy. --best, kevin [kzollman][talk] 04:06, 31 May 2007 (UTC)[reply]