# A Game-Theoretic Intelligent Agent for the Board Game Football Strategy

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2 version of the matrix is included in Figure 1. In the actual chart, there are 20 offensive plays and 10 defensive plays, for a total of 200 entries. FUMBLE FUMBLE FUMBLE Figure 1: A subsection of a Football Strategy chart In each play of the game, the offensive player and defensive player each secretly and simultaneously choose a letter or number. The intersection of those choices in the chart determines the result of the play. In most cases, the chart gives the gain (or loss) of yardage as the result of the play. For example, the combination of offense play 1 and defense play D (we will abbreviate such a combination as 1D ) is a gain of one yard. In some cases, other results can happen-- penalties, or as in the case of play 2B, a turnover, giving the other team the ball. The normal rules of American football apply (for scoring, first downs, et cetera.) While some combinations of offensive and defensive plays lead to the same outcome (for example, plays 4B and 3D above both result in a gain of two yards), the chart has over 60 unique outcomes, leading to a branching factor that is too large to be addressed with traditional methods. By way of comparison, Chess has an average branching factor of about 35. Game Theoretic Approach In general, the chart given above can be seen as a zero-sum normal form game, in the game-theoretic sense. In these games, a mixed strategy is a probability that each player will choose one of each available options. A Nash Equilibrium (Nash 1950) in this case is a mixed strategy that cannot improve any player s utility, given the other player s strategy. Von Neuman has shown (von Neumann and Morgenstern 1944) that a mixed strategy Nash Equilibrium exists for all zero-sum normal form games. The mixed strategy for each player can be computed using linear programming (for example, see Mendelson 2004.) We have developed an agent that uses these techniques to play the game of Football Strategy. The game chart gives the basic utility function measured in yards gained by the offensive player. The agent then modifies the values in the chart by adding (or subtracting) values to each table entry based on how the play affects the overall game. The new weighted yardage table is then solved using linear programming to generate a mixed strategy. The resulting mixed strategy will give the agent a probability of choosing each offensive (or defensive) play that will maximize (or minimize) the expected performance of the offensive player. Table 1 shows a sample mixed strategy for an offensive player, and Table 2 a sample mixed strategy for the defensive player. Both tables show the same situation (first and ten, on the offense s twenty yard line.) Lownumbered offensive plays tend to be running plays, and high numbers tend to be passing plays. Low-lettered defensive plays tend to be defenses against runs, and high-lettered defensive plays tend to be defenses against passes. In this case, the intersection of the highest probability offensive play call and the highest probability defensive call is 4E, which would result in a gain of five yards for the offense. However, most of the other defensive play calls that have positive probability (A,B,C,D,and H) give lower gains against the offensive play call of 4, sometimes even resulting in losses of yardage for the offense. Offensive play number Probability of Play Call 1 0% 2.29% % % % 6 0% 7 0% 8 0% % 10 0% 11 0% 12 0% % 14 0% 15 0% 16 0% % % 19 0% 20 0% Table 1: An example of an offensive mixed strategy It is interesting to note that there are several play calls that have zero probability-- they will never be called. This result indicates that the solver has determined that for the given game situation (defined by a combination of yard line, down, distance to a first down, score, and time remaining in the game), those play calls are dominated by others. It is the case that each offensive and defensive play has a game situation in which it will be called with a nonzero probability.

5 Future Work References There are several possible avenues of future improvement that can be added to the agent. We would like to generate the values of the five constants used to adjust the values placed in the game matrix in a better way. Right now we have just used trial-and-error in playing against humans. We would instead like to make these values more rooted in the underlying mathematical principles of the game. We would like to integrate calling time-outs into the utility function, especially towards the end of each half of the game. Currently, time-outs are called outside of the game matrix decision making, using a set of pre-defined rules. Incorporating them into the utility matrix would allow for more strategic decision making by the agent. We would like to explore ways in which the agent can plan a sequence of downs to realize short-term goals. For example, a second and short situation- where it is second down and very few yards are needed to obtain a first downis often a good time to attempt a risky play like a long pass. If it is successful, there is a large gain of yardage. If it is unsuccessful, it is likely that the short yardage required for a first down can be gained on the next two plays. Currently, the agent as constructed merely sees a high probability for obtaining a first down and chooses that over the higher-potential play. Humans tend to play Football Strategy by predicting tendencies of play calls made by the opponent. It would be interesting to explore whether a learning element could be added to the game to allow the agent to make similar decisions. A similar game to Football Strategy exists, called Paydirt (BGG 2015b). This game uses a chart of play calls similar to the one found in Football Strategy, but the charts represent actual historical teams. The intersection of an offensive and defensive play call is resolved probabilistically based on the historical team s abilities. The underlying ideas behind our current agent should translate well to a program playing this similar game. We would like to extend this idea of exploiting the underlying mathematical structure of a game to other games. While there are not many games that are so directly related to normal-form games, many other games use concepts from probability, graph theory, or auctions that can be exploited in similar ways. BGG. 2015a. Football Strategy. Available online at BGG 2015b. Paydirt. Available online at BPA Boardgame Players Association Football Strategy Rankings. Available online at BPA Boardgame Players Association. Available online at Hall L.; Gordon A.; James R.; and Newall L A Lightweight Rule-Based AI Engine for Mobile Games. In Proceedings of the 2004 ACM SIGCHI International Conference on Advances in Computer Entertainment Technology, New York, NY: Association of Computing Machinery. Heyden C Implementing a Computer Player in Carcassone. Master s Thesis. Department of Knowledge Engineering, Maastricht University, Maastricht, Netherlands. Johnansson, S On Using Multi-agent Systems in Playing Board Games. In Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems, New York, NY: Association of Computing Machinery Mendelson, E Introducing Game Theory and its Applications. Boca Raton FL:CRC Press. Nash, J Equilibrium Pointes in n-person Games. Proceedings of the National Academy of Sciences, 36(1): Schadd, M,.; Winands M.; Bergsma M; and Uiterwijk J Solving Fanorona. ICGA Journal, 30(1): Schadd, M; and Winands M Quiescence Search for Stratego. In Proceedings of the 21 st Benelux Conference on Artificial Intelligence. Eindhoven, the Netherlands. Thomas, R Real-time Decision Making for Adversarial Environments Using a Plan-based Heuristic. PhD diss., Department of Computer Science, Northwestern University, Evanston IL. von Neumann, J., and Morgenstern, O Theory of Games and Economic Behavior. Princeton University Press. Acknowledgements We would like to thank Julide Yazar for an invaluable education into the basics of Game Theory. We would also like to thank Bruce Reiff for being our resident Football Strategy expert and experimental subject.

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