game theory tree latex two stage draw only one outcome
Last week instead of writing a post on game theory I ended up writing some code related to game theory. The most commonly described games are 2×2 matrix games, and in an effort to make game theory easier to understand, I developed a website that can solve any 2×2 matrix game.
Here is a thumbnail preview to the solver:
http://mindyourdecisions.com/GameSolver.html
The solver is presented as a game between Rose and Colin. I came across this pragmatic naming convention in the book Game Theory and Strategy by Philip D. Straffin. The player Rose selects between the rows "Up" and "Down" while the player Colin selects one of the columns "Left" or "Right."
The rest of the solver is self-explanatory. Just enter the payoffs and the program will automatically solve for the game's Nash equilibrium in pure and mixed strategies. You can edit the matrix payouts and the solver will update the results, or you might want to start over and reset the payoffs by clicking the button.
In this post, I'll illustrate how the solver operates for the most famous 2×2 matrix games.
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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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A Video Explanation
For those that prefer, I have also made a video that previews how the solver works.
The Game Theory Solver: Solve Any 2×2 Matrix Game Automatically
As ever, you can view more videos on math and game theory on my YouTube channel.
The Prisoner's Dilemma
We'll start out with the most famous problem in game theory. Although this game is logically counter-intuitive, it is mathematically one of the easiest examples to solve!
Let's say players get 2 when both cooperate, 3 when only one defects or 0 when only one cooperates, and 1 when both defect. We enter those payouts.
And the solver identifies there is a unique Nash equilibrium where both defect and end up with 1.
Matching Pennies
In matching pennies, each of two players shows "heads" or "tails." Rose gets +1 if the two choices match and -1 if the two choices do not match. Colin gets the opposite payouts.
If we think about the labels "Up" = "Left" = "Heads", and "Right" = "Down" = "Tails," then we can write the payouts in our solver.
The 2×2 matrix has Rose getting +1 in the upper left and lower right entries, -1 in the other two, and Colin getting the opposite payout of Rose. We enter those payouts.
Instantly the solver identifies there is no Nash equilibrium in pure strategies and it also solves for the unique Nash equilibrium in mixed strategies.
Battle of The Sexes
We'll skip the narration on this game. The payouts are (3, 2) is the payout for (Up, Left), (2, 3) is the payout for (Down, Right), and the rest are 0's, which we input.
The solver explains there are 2 pure Nash equilibrium and a unique mixed strategy.
Game of Chicken
This is a similar game to the battle of the sexes mathematically. We'll enter the payouts.
The solver again identifies the two pure strategy Nash equilibrium and the unique mixed strategy equilibrium.
(There are some rounding issues as the solver works numerically. So you should recognize the mixed strategies are 1/3 and 2/3 with an expected payout of 4.7).
Anything Goes
Let's say that Rose and Colin get 2 a piece no matter what each chooses.
Now everything is a pure strategy Nash equilibrium and there are infinitely many mixed strategy Nash equilibrium too, both identified by the solver!
Anything Goes for one player
What happens when one player has a dominant strategy, but the other player is indifferent between two choices? The player with the dominant strategy picks it, and the other player can choose any level of mixing.
Inputting such a game:
The solver identifies Rose has a dominant strategy of "Up" and Colin can mix Left and Right in any proportion.
Game with exactly 2 equilibria
Let's say (Up, Left) has the payout (1, 1) and everything else has a payout of 0.
The solver explains there are exactly 2 pure strategy Nash equilibria. The interesting part of this game is there are an even number of equilibria, and most games have an odd number.
The Game Theory Solver for 2×2 Games
I hope you find the 2×2 game theory solver useful. Here is a link to the website. I hope you'll find it useful to share with your economics teacher (or teachers to your students).
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Source: https://mindyourdecisions.com/blog/2014/09/30/game-theory-tuesdays-2x2-matrix-game-solver/
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