|WikiProject Game theory||(Rated Start-class, Top-importance)|
Merge with payoff matrix
This merge seems valid but "Normal Form Game" seems a more appropriate title than "Payoff Matrix". Suggest merging payoff game into this article.
why merge payoff matrix into normal form game? A payoff matrix is indeed a tool that is used in game theory, but not exclusively. It is also used in decision theory, where there may be a payoff matrix with just one decision maker: his possible decisions are on one axis and random events on the other: so, it is a payoff matrix, but not like what is described in this article. I suggest either making separate articles, or making a general aticle under the title "payoff matrix", having "normal form game" as a particular case of pyoff matrices (since it seems to me the latter is a subcategory of the former - I'm not a specialist though, so I might be wrong).184.108.40.206 (talk) 13:27, 17 June 2008 (UTC)
infinite / continuous game
I took out the bit about continuous games. First, those generally aren't represented as matrices. (I suppose it could be a tensor, but that's certainly not typical), so it doesn't apply to normal form games. Second, it's not true that functional analysis is necessary to solve continuous games.Cretog8 (talk) 06:25, 27 June 2008 (UTC)
- To my knowledge, a “normal form” game refers to the representation of a game where both players make their moves simultaneously in one turn (although mathematically extensive form and normal form games are equivalent.) The formulation of continuous games given on that page refers to games where each player makes his move simultaneously, as in the discrete case of the normal form game, making the relation between the two quite natural.
- Matrices are only used to index payoff functions of two-player discrete (normal form) games; they are not needed to define normal form games. I would imagine tensors would be used to describe the payoff functions of 3+ player discrete games, although this is hardly ever used to my knowledge. The pay-off functions of continuous games will in general be described by multivariate functions.
- by the definitions given on continuous game and on this page, normal form games (as well as the equivalent extensive form) can formally be considered a subset of continuous games as any finite set of strategies can be considered a compact metric space under the discrete metric, and any function over a discrete metric space will be continuous.
- Some continuous games do require the use of functional analysis to find mixed-strategy Nash equilibria; for example for games with pure strategy set [0,1] for each player, we sometimes need to consider the set of all possible probability measures over [0,1] to find a mixed strategy equilibrium. --220.127.116.11 (talk) 23:22, 27 June 2008 (UTC)
- This might take some hashing out.
- The continuous game article is effectively brand-new, so it might need a good bit of review itself before I'm willing to take it as the point of reference.
- To my understanding, the defining feature of a continuous game is the strategy space, not the number of players, simultaneity, or payoff functions.
- "Normal form game" is to "finite simultaneous game" as "extensive form game" is to "finite dynamic game". They get conflated, but the "form" is about the representation, not strictly about the kind of game. This article shows how "normal form" can be applied to dynamic games (which I suspect is more confusing than helpful, myself, but...). So, since it is not practical to represent a continuous-strategy game in matrix form, I don't see it as being useful to describe them as "normal form". They may indeed be simultaneous.
- Multiple matrices can (and are) used for payoffs in games of more than 2 players. 3 players usually isn't too bad, more gets pretty ugly.
- I'm not sure about your point of finite games being a subset of continuous games. It may be that you're formally right, but is it helpful?
- Sorry. Yes, some continuous game undoubtedly require functional analysis. The text I removed said that they required functional analysis in general, while many don't. Cretog8 (talk) 00:09, 28 June 2008 (UTC)
- This might take some hashing out.
- Possibly the problem is the focus on articles for normal form game and extensive form game. Really, the bulk of the material should probably be in simultaneous game and sequential game, instead, with the "form" articles limited to details of representation. Just a thought, which should probably be brought up at the game theory project page before following through. Cretog8 (talk) 00:14, 28 June 2008 (UTC)
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The Example states: "...if player 1 plays top and player 2 plays left...". I'm new to game theory and this is the first time I've read this page. What is "top"? Are these the bottom, right, top and left of a rectangle? What value does top have? --> 4? What values do the other side have, if they are sides? etc, etc.
After reading this page I'm worse off than before I read it. Like many wiki technical articles, it's been written by someone with an in depth knowledge but lack of ability in passing this knowledge on. —Preceding unsigned comment added by 22.214.171.124 (talk) 14:58, 6 May 2011 (UTC)