The duality gap for two-team zero-sum games

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k = 1, such zero-sum games possess a unique value, independent of order of play. However, this fails for all k > 1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. We show that the gap equals 2(1−2^(1−k)) for m = 2 and 2(1−m^(−(1−o(1))k)) for m > 2, with m being the size of the action space of each player. Extensions hold also for different-size teams and players with various-size action spaces. We further study the effect of exchanging order of commitment among individual players (not only among the entire teams). The class of two-team zero-sum games is motivated from the weak selection model of evolution, and from considering teams such as firms in which independent players (ideally) have shared utility

The Duality Gap for Two-Team Zero-Sum Games

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k=1, such zero-sum games possess a unique value, independent of order of play, due to the von Neumann minimax theorem. However, this fails for all k>1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. In our main result we show that the gap equals 2(1-2^{1-k}) for m=2 and 2(1-m^{-(1-o(1))k}) for m>2, with m being the size of the action space of each player. At a finer level, the cost to a team of individual players acting independently while the opposition employs joint randomness is 1-2^{1-k} for k=2, and 1-m^{-(1-o(1))k} for m>2. This class of multiplayer games, apart from being a natural bridge between two-player zero-sum games and general multiplayer games, is motivated from Biology (the weak selection model of evolution) and Economics (players with shared utility but poor coordination)

The duality gap for two-team zero-sum games

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k = 1, such zero-sum games possess a unique value, independent of order of play. However, this fails for all k > 1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. We show that the gap equals 2(1−2^(1−k)) for m = 2 and 2(1−m^(−(1−o(1))k)) for m > 2, with m being the size of the action space of each player. Extensions hold also for different-size teams and players with various-size action spaces. We further study the effect of exchanging order of commitment among individual players (not only among the entire teams). The class of two-team zero-sum games is motivated from the weak selection model of evolution, and from considering teams such as firms in which independent players (ideally) have shared utility

Mapping the landscape of metabolic goals of a cell

Genome-scale flux balance models of metabolism provide testable predictions of all metabolic rates in an organism, by assuming that the cell is optimizing a metabolic goal known as the objective function. We introduce an efficient inverse flux balance analysis (invFBA) approach, based on linear programming duality, to characterize the space of possible objective functions compatible with measured fluxes. After testing our algorithm on simulated E. coli data and time-dependent S. oneidensis fluxes inferred from gene expression, we apply our inverse approach to flux measurements in long-term evolved E. coli strains, revealing objective functions that provide insight into metabolic adaptation trajectories.MURI W911NF-12-1-0390 - Army Research Office (US); MURI W911NF-12-1-0390 - Army Research Office (US); 5R01GM089978-02 - National Institutes of Health (US); IIS-1237022 - National Science Foundation (US); DE-SC0012627 - U.S. Department of Energy; HR0011-15-C-0091 - Defense Sciences Office, DARPA; National Institutes of Health; R01GM103502; 5R01DE024468; 1457695 - National Science Foundatio

Continuous guts poker and numerical optimization of generalized recursive games

We study a type of generalized recursive game introduced by Castronova, Chen, and Zumbrun featuring increasing stakes, with an emphasis on continuous guts poker and $1$ v. $n$ coalitions. Our main results are to develop practical numerical algorithms with rigorous underlying theory for the approximation of optimal mutiplayer strategies, and to use these to obtain a number of interesting observations about guts. Outcomes are a striking 2-strategy optimum for $n$-player coalitions, with asymptotic advantage approximately $16\%$; convergence of Fictitious Play to symmetric Nash equilibrium; and a malevolent interactive $n$-player "bot" for demonstration

Futures Market: Contractual Arrangement to Restrain Moral Hazard in Teams

Holmstrom (1982) argues that a principal is required to restrain moral hazard in a team: wasting output in a certain state is required to enforce efficient effort, and the principal is a commitment device for such enforcement. Under competition in commodity and team-formation markets, I extend his model a la Prescott and Townsend (1984) to show that competitive contracts can exploit the futures market to transfer output across states instead of wasting it. Thus, the futures market replaces the role of principals. An important feature of such contracts is exclusiveness: private access to the the futures market by team members is not allowed. The duality of linear programming is exploited to characterize a market environment and the contractual agreements for efficiency.