The Gritty Stuff

This week is going to be a bit dry, but it will pay off in the future. Dexter mentioned in the comments last week that it would be nice for me to start using payoff matrix. And so this week I am going to lay out some tools used in game theory that will be used to make it easier to notate and visualize the concepts as we talk about them.

First up is the payoff matrix that Dex mentioned. This is just a way to neatly arrange a problem and make it easier to see things like dominated strategies and the such. For this example let’s say there are two players, i and j. They both have two choices, yes or no. If both players say yes, they both have a payoff of 2. If both players say no, then they both have a payoff of 0. If a player say yes, and the other player says no, then the player who says yes gets a payoff of 1, and the player says no gets a payoff of 3. Now that’s a lot of text, and as these problems get even larger, it will be hard to sift through all of it. This is where the payoff matrix comes in.

i (left) / j (top) Yes No
Yes 2, 2 1, 3
No 3, 1 0, 0

The one important part to remember is that the value on the left corresponds with the player whose strategies are on the left. So the three, in the bottom left, corresponds to the No of the player i. 

Now to get through the most boring part, notation. Now don’t let that term scare you off, this isn’t complex notation. It is really just more of a shorthand that makes it easier as the problems get larger. So I will dive right in:

Players                  i, j               Variables that stand in for specific players

Strategies            si                 A particular strategy of player i

Si                 All possible strategies of player i

                                       s-i              Strategy choice for all players but player i

        Payoffs                U(si, sj)      Think of this like a point being plugged into the function of                                                               U, the output being the payoff of the two players

                                      Ui(si, sj)      Same as above, except the output will be the payoff of                                                                         player i


There, now that is over with, and if you ever are confused by notation I used just refer back to this page. Just to make sure you understand both concepts, I will combine them for a few examples that can be used to check your understanding. Si = {yes, no}. The U(yes, no) = (1, 3). The Ui(no, yes) = 3. If you have any questions on that just ask in the comments and I will do my best to clear it up.

Now to add some joy to what would be an otherwise fairly dull blog post this week, I am going to pose a fun challenge. I will give a payoff matrix, and see if you can determine the best choice for player i by deletion of dominated strategies. If so, say which and why in the comments.

i, j a b c d
A -2, 2 1, 4 3, 1 5, 0
B 2, -1 3, 1 5, 4 3, -2
C -4, 3 0, 0 1, 1 -2, 5
D -1, 4 1, 3 2, 2 4, -1

I hoped you learned something, thanks for reading.

For anyone who wants to follow their own interest in game theory, here are a few resources:



2 thoughts on “The Gritty Stuff

  1. dexcoengilbert

    Thanks for taking my suggestion! It was definitely helpful to see the payoff matrices right in front of me.


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