<p>If both players have a 1-dimensional strategy space, then the graph of the potential function is a &quot;surface&quot; over the 2-dimensional Cartesian product of these strategy spaces. This of course generalizes to higher dimensions, but then it can no longer be visualized for a geometric intuition.</p>
<p>However, a game without a potential function doesn't correspond to a &quot;surface without extrema&quot;, it just corresponds to no such surface existing.</p>

CC BY-SA 4.0
2023-01-03T16:13:07.323Z
Is there a meaningful relationship between this notion of a potential function and the MV Calculus notion of a potential function (basically the inverse of the gradient operator), or is this an artificially constructed similarity?
user10478
CC BY-SA 4.0
2023-01-04T14:28:33.327Z
@user10478, as far as I remember there is some relationship that is more than just a similarity, but I don't know the details. Maybe try asking this question at Math StackExchange.
VARulle
CC BY-SA 4.0
2023-01-03T13:34:31.217Z
2023-01-03T13:34:31.217Z
VARulle
<p>One geometric interpretation of (at least one term of) the potential function I've come across is as the Riemann-approximated area under an individual player's cost as a function of the number of players...</p>
<p><a href="https://i.sstatic.net/yhfZH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yhfZH.png" alt="enter image description here" /></a></p>
<p>However, the potential function is not actually a function of the number of player, but of their strategies.  Is there a geometric interpretation of the potential function for, i.e., a game with two players as a three-dimensional surface, where the input dimensions are the strategies of the two players or something like that, and the potential function's extrema (which correspond to Nash Equilibria) can be seen?  Presumably, a game without a potential function would correspond to a surface without extrema?</p>

<mathematical-economics><game-theory><nash-equilibrium><mechanism-design><potential-game>
Geometric Interpretation of the Potential Function of a Game
user10478
# Geometric Interpretation of the Potential Function of a Game

(Asked by: username_0 on 2023-01-03)

One geometric interpretation of (at least one term of) the potential function I've come across is as the Riemann-approximated area under an individual player's cost as a function of the number of players...

[![enter image description here](https://i.sstatic.net/yhfZH.png)](https://i.sstatic.net/yhfZH.png)

However, the potential function is not actually a function of the number of player, but of their strategies. Is there a geometric interpretation of the potential function for, i.e., a game with two players as a three-dimensional surface, where the input dimensions are the strategies of the two players or something like that, and the potential function's extrema (which correspond to Nash Equilibria) can be seen? Presumably, a game without a potential function would correspond to a surface without extrema?

**Tags:** mathematical-economics, game-theory, nash-equilibrium, mechanism-design, potential-game

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## 1 answers

### Answer by username_1 on 2023-01-03. Score: 2 | Accepted answer

If both players have a 1-dimensional strategy space, then the graph of the potential function is a "surface" over the 2-dimensional Cartesian product of these strategy spaces. This of course generalizes to higher dimensions, but then it can no longer be visualized for a geometric intuition.

However, a game without a potential function doesn't correspond to a "surface without extrema", it just corresponds to no such surface existing.

> **Comments:**
> **username_0**: Is there a meaningful relationship between this notion of a potential function and the MV Calculus notion of a potential function (basically the inverse of the gradient operator), or is this an artificially constructed similarity?  
> **username_1**: @user10478, as far as I remember there is some relationship that is more than just a similarity, but I don't know the details. Maybe try asking this question at Math StackExchange.  

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