Talk:Strategy (game theory)

Non-strategic examples

The roulette and bet-hedging stock trading strategies are not game theoretical and should go to some other article. Pete.Hurd 18:51, 17 September 2006 (UTC)

Just got rid of them. Any important information is in other articles already. Cretog8 (talk) 09:10, 1 June 2008 (UTC)

Tic Tac Toe

Just removed this: "The number of "moves" in a Tic Tac Toe game is 4 or 5, depending on whether you start or not, and considering that neither player can skip a turn; while the actual number of "strategies" is over 6 trillion."

I don't know the number of strategies for tic-tac-toe, but it's certainly not 6 trillion. To give an upper bound on the number of strategies, you can do this: There are 9 spots in tic-tac-toe and each of them can be in one of 3 states: {blank, X, O}. So, there are 3^9 = 19,683 possible states for the whole board. A strategy could be defined by a rule for what to do in every state, so 19,683 strategies would be exhaustive. (Actually that number's way too high, considering that most of the board states would be impossible in a real game or wouldn't correspond to your move, but...)

Actually, a stategy has to tell you what to DO in each state (for the mathematically inclined, a stategy is a function from the set of possible board states to the set of possible moves). For each state you have up to nine squares you can choose to move in, which gives an actual upper bound of 9^19683 stategies, which is, umm, really really big. Of course not all 19683 can occur in a real game (namely those where one or the other player would have already won, or where the numbers of X squares and O squares differ by more than 1), and in most states some squares will be occupied, but I still find "over 6 trillion" quite believable. My calculations give me an upper bound of 6*10³¹⁴⁷ (still not taking into account states where the game has already been won, but taking into account alternation of turns). In fact, my calculations show 10⁹¹⁴ as a very, very crude lower bound (considering only the first five moves, when it's impossible for anyone to have won yet), making 6 trillion seem like a heavy understimate. Symmetry considerations might crunch the number of effectively different strategies down further... (In the first turn, for example, your only real options are "corner", "edge", and "center" - which corner or which edge you play doesn't matter.) -- Milo —Preceding unsigned comment added by 195.241.9.38 (talk) 18:05, 25 July 2008 (UTC)

If allowing for mixed strategies, then there's an infinite number, but that's something else. Cretog8 (talk) 09:04, 1 June 2008 (UTC)

From the Tic-Tac-Toe article: Ignoring the sequence of Xs and Os, and after eliminating symmetrical outcomes (ie. rotations and/or reflections of other outcomes), there are only 138 unique outcomes. Assuming once again that X makes the first move every time: 91 unique outcomes are won by (X) 21 won by (X) after 5 moves 58 won by (X) after 7 moves 12 won by (X) after 9 moves 44 unique outcomes are won by (O) 21 won by (O) after 6 moves 23 won by (O) after 8 moves 3 unique outcomes are drawn

It sounded very ridiculous that there would be 9^19683, and the Tic-Tac-Toe article explains it well. Aeonoris (talk) 17:53, 29 July 2009 (UTC)

merge, planetmath and stuff

This article is a merge of its previous incarnation, mixed strategy and pure strategy. I've removed references to PlanetMath, because while there is a little bit from PlanetMath, it is very little. If this lack-of-citation is inappropriate, I'm happy to be corrected.

equilibrium

Currently, the only mixed-strategy equilibrium example given is a link to rock-paper-scissors, but neither page gives any detail. More examples would be good, particularly of stable mixed strategy equilibria. One specific example that would be very good is some kind of tacit collusion (say, it would seem that small groups of service stations price petrol below the profit maximising monopoly price but above the economic equilibrium for perfect competition, and where can one learn how to explicitly calculate this eventual price?). Cesiumfrog (talk) 05:48, 3 June 2008 (UTC)

Wow, I'm surprised at Rock paper scissors. The only concrete game theory there jumps right to a biological application. I've replaced the link to RPS with one to Matching Pennies. I hate using MP as an example, because I've never heard of it outside of its pedagogical use in game theory. Maybe if/when equilibrium stuff gets added to RPS the link can get switched back.Cretog8 (talk) 11:21, 3 June 2008 (UTC)

One of the things I wonder about this article is how much it should go into equilibrium. After all, a strategy (pure or mixed) can be used out of equilibrium. So, I'm fuzzy on how many equilibrium examples should be given. I also wonder if A disputed meaning is too equilibrium-focused. Cretog8 (talk) 11:25, 3 June 2008 (UTC)

disputed meaning

I just re-worked the "disputed meaning" section a bit. I hoped to clarify it, and hopefully I have some, but I think the wording and flow could still use some more clarification.

Also, there's a gap in that section, but I'm not sure how to resolve it without getting into WP:OR. Mixed strategies can occur in different kinds of equilibrium, or out of equilibrium. The problems with mixed strategies apply in any case. However, the solutions described in that section apply almost entirely to Nash equilibrium. Not ure how to get at it, but hopefully someone else can find a non-OR way to fill in the gap.Cretog8 (talk) 21:41, 24 June 2008 (UTC)

Wording of introduction

I find the wording of the introduction contradictory:

In game theory, a player's strategy is any of the options which he or she can choose

[...]

A strategy [...] is a complete algorithm for playing the game,

In my understanding, there is a difference between an option (chosen from a set of possible options) and an algorithm (that is designed to pick such option. — Preceding unsigned comment added by 2A02:8108:40C0:ED5:9142:E987:A3C2:7493 (talk) 15:48, 16 November 2018 (UTC)

Confusion of purpose

The section "dissing" mixed strategies seems to be out of place -- it should be toned down or removed. Obviously there isn't anything "problematic" about mixed strategies that win. The problem in the article is that the role of game theory as a battle of rational actors is being confused with behavioral economics a la Kahneman, etc.

These are two different sets of questions, for two different purposes. One has to do with winning against all comers, the other has to do with explaining human behavior. Needless to say, a poker player will want to to understand the latter but practice the former -- despite being human himself.

If this "social slide" really must be made in this particular article, it should be framed in such a way to keep the two Fragestellungen separate from each other.

2001:67C:10EC:578F:8000:0:0:46C (talk) 13:00, 9 July 2020 (UTC)

Still lots of inconsistencies

At first reading some contradictory or misleading definitions are obvious. This is not appropriate for a technical article like this, is it? Who feels a calling to correct this?

A strategy is any of the options which a player chooses. / A strategy is a complete algorithm. / A strategy is a list of directions.

A strategy set is the set of strategies that are available for a player.

A strategy profile is a set of strategies. / A strategy profile is an (ordered) set of strategy sets.

--Hpstricker (talk) 10:53, 3 March 2021 (UTC)

"Strategia (teoria dei giochi)" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Strategia (teoria dei giochi) and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 12#Strategia (teoria dei giochi) until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (talk) 02:10, 12 February 2022 (UTC)

Plagiarism?

In section "Strategy with perfect recall", it says "Figure [2] describes this game.". There is no Figure 2 on this page and the style of the reference suggests that parts of or the whole paragraph was copied from some paper. 81.227.162.206 (talk) 08:53, 22 May 2023 (UTC)