Balancing Zero-Sum Games with One Variable per Strategy

Albert Julius Liu, Steve Marschner


A key challenge in game design is achieving balance between the strategies available to the players. Traditionally this has been done through playtesting, with its difficult requirements of time, labor, and interpretation of results. To make it quicker and easier to balance games, we propose a game-theoretic approach that automatically balances strategies based on a mathematical model of the game.

Specifically, we model the balance problem as modifying a zero-sum game, using one variable per strategy, so that every strategy has an incentive to be employed. We begin with a special case where these variables affect player payoffs multiplicatively, and show that the simple Sinkhorn-Knopp algorithm can be used to balance the game. We then proceed to analyze the more general case where the variables have a monotonic effect on payoffs, and show that it is amenable to standard optimization methods. We give examples inspired by well-known game series including Pokémon and Warhammer 40,000.


AIIDE 2017.

Direct link to PDF (1.6 MB).


The one-up game (313 KB).

In this appendix we analyze the "one-up" game. This demonstrates the technique of analyzing a continuous strategy set by discretizing it and running it through our algorithm. We also analyze the effects of restricting one player's strategy set.

Source code



    author = {Albert Julius Liu and Steve Marschner},
    title = {Balancing Zero-Sum Games with One Variable per Strategy},
    booktitle = {AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment},
    month = oct,
    year = {2017},
    url = {}