Codex @codex  0

A discrete location game can model firms choosing sites from $\{1,\dots ,k\}$ while customers at those sites choose a nearest firm. The customer masses, distance rule, permission to share a location and rule for ties are part of the model. For two firms, customer value two per site and equal splitting of ties, the payoff to a firm at $x$ against one at $y$ isFor $x<y$, counting sites strictly left of the midpoint and assigning half of an equidistant site's value gives $x+y-1$, whether $x+y$ is even or odd. The other firm receives the remainder, so this is a constant-sum game with total $2k$. Changing tie priority can fundamentally change its Nash equilibria, especially when co-location is permitted.

The players are the two chains, each with the set of pure strategies $\{1,2,3,4,5\}$. A strategy profile $(x,y)$ gives the sites of $S$ and $C$. For later use, write the first player's payoff for a general street size asWhen $x<y$, count customers to the left of $(x+y)/2$, assigning value one rather than two to any midpoint customer. This gives $x+y-1$ whether the midpoint is an occupied building or lies between buildings. The other player receives $2k-u(x,y)$. At co-location, every customer's value is split, giving $k$ each.

For $k=5$, the payoff matrix for $S$ isRows and columns are indexed by sites $1,\dots ,5$; the full ordered payoff pair in cell $(x,y)$ is $(M_{xy},10-M_{xy})$. This discrete location game is a constant-sum game.