Talk:Normal-form game

Merge with payoff matrix

Please discuss the possible merger of these two articles here. Thanks! --best, kevin ···Kzollman | Talk··· 21:15, 8 October 2005 (UTC)

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).62.169.208.134 (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)

Cretog8,

• 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. --67.86.29.2 (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)

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)

I see. I did not realize that "form" explicitly referred to representation. Ultimately this is a question of how the newborn article on continuous games will fit into the larger Wikipedia framework on game theory. 67.86.29.2 (talk) 02:04, 28 June 2008 (UTC)

Comprehensive Example?

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 82.41.42.125 (talk) 14:58, 6 May 2011 (UTC)

Strategic vs normal form game

I think there should be a page "strategic-form game" that links to this page. Many authors use the term strategic form instead of normal form. — Preceding unsigned comment added by 140.112.177.67 (talk) 07:02, 6 November 2018 (UTC)

"Gioco in forma normale" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Gioco in forma normale and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 12#Gioco in forma normale until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (talk) 02:11, 12 February 2022 (UTC)

"Payoff matrix" is used without definition

The term is used first in « The topological space of games with related payoff matrices can also be mapped, with adjacent games having the most similar matrices. » and then in « The payoff matrix facilitates elimination of dominated strategies, and it is usually used to illustrate this concept. » And those sentences are not accompanied by any definition. One may think that the definition is trivial but it is not: for a two-player zero-sum game specifying only one real number for each pair of moves of the 2 players is enough thus the payoff matrix can be real-valued, but when the sum is not 0 (or fixed) one must give 2 values for each entry of the matrix, which is thus not a real-valued matrix -unless one introduces a coding of pairs of real numbers but that is not standard practice.

In the first sentence quoted the expression « the most similar matrices » is similarly undefined. And more generally i find the presentation of this page rather imprecise and in some places wordy. For instance « graphical » in the first sentence of this page links to the "Graph" wiki page, thus "graphical" is implicitly defined as "relating to or in terms of graphs". But "graphical" could well be interpreted in the more habitual sense of "relating to or in the form of a diagram or an image"; and an extensive form of a game is not just a graph so if it is "graphical" then this term rather means "in the form of a diagram". Plm203 (talk) 09:37, 23 September 2023 (UTC)