# Illustrating difficulty to find dominant strategies: series of simple one-shot two-players games

For a game theory class I am prepping, I am looking for simple mathematical puzzles that can easily be presented as one-shot 2-players simultaneous games in which one of the players has a dominant strategy. The puzzle is to find the dominant strategy.

My goal is to interactively illustrate how "games are not equal in the face of dominant strategies": Although standard game theory predicts that players will play dominant strategies when they have one, dominant strategies can be more or less easy to identify in different games, which may lead players to play dominant strategies more often in some games than others.

I would like to do so by putting students in the shoes of the player who has a dominant strategy in a series of games in which the dominant strategy is increasingly hard to find.

Note that the game of Nim is excluded because it is neither one-shot nor simultaneous. The games could for example (but do not have to) look like this:

The other player will choose and integer $$y$$ satisfying property $$P_1$$ (e.g., $$y$$ is contained in some subset of integers). You must yourself choose an integer $$x$$ satisfying property $$P_2$$. You win if $$f(x,y)$$ (e.g., $$x+ y$$) satisfies property $$P_3$$.

My question is: can anyone provide a series of games that would do the job?

Desirable qualities of the series are:

• Ideally, the series should contain between three and four games.
• Games should be easy to explain.
• Games should be "truly interactive" in the sense that the other player can actually influence the outcome (no "The other player will choose $$y$$. You must choose positive integer $$x$$. You win if $$31^3+33^3+x^3 = 76^3$$.")
• Games should be as "entertaining" as possible (sorry, I know this is subjective).
• The progression in the difficulty to find dominant strategies should ideally come from dominant strategies becoming more and more "clever", as opposed to checking an increasing number of "cases" (sorry, I know this too is subjective).
• I should be able to explain why the dominant strategy is what it is in class relatively quickly (I.e., ideally dominant strategies should be of the kind "hard to find, but easy to understand once you know what it is")

Start with the following matrix. $$\begin{matrix}1 && 0 && 0\\ 1 && 1 && 0\\ 1 && 1 && 1 \end{matrix}$$
You will choose a $$1$$ in the matrix and change it to a $$0$$. The other player will choose a $$0$$ and change it to a $$1$$. Your goal is for the resulting matrix to have an even determinant. The other player's goal is for the resulting matrix to have an odd determinant.
Change the $$1$$ in the bottom left corner to a $$0$$.