Researcher Solves 60-Year-Old Game Theory Problem
Researcher solves 60-year-old game theory problem. To comprehend how autonomous vehicles can navigate intricate road scenarios, researchers often employ game theory, a mathematical model depicting the strategic behavior of rational agents to achieve their objectives.
Dejan Milutinovic, a professor of electrical and computer engineering at UC Santa Cruz, has focused on differential games, a complex subset of game theory that deals with moving players. One such game is the wall pursuit game, a simple model involving a faster pursuer aiming to catch a slower evader constrained to a wall path.
Researcher solves 60-year-old game theory problem: IEEE Transactions on Automatic Control
For nearly 60 years, a predicament persisted in this game, where a range of positions seemed to lack an optimal solution. However, Milutinovic and his team recently proved, in a paper published in IEEE Transactions on Automatic Control, that this dilemma isn’t valid. They introduced a novel analytical approach that demonstrates a deterministic solution always exists for the wall pursuit game. This finding could help resolve similar challenges in differential games and enhance reasoning about autonomous systems, like driverless vehicles.
So, Game theory finds application in various fields, encompassing economics, politics, computer science, and engineering, to comprehensively understand behavior. The Nash equilibrium by mathematician John Nash signifies optimal strategies minimizing regret for all game players. This concept extends to the wall pursuit game, where rational players adopt their equilibrium strategy to avoid increased regret.
The classical analysis for this game fails for a specific set of positions, known as a singular surface, leading to the acceptance of the dilemma. However, Milutinovic and colleagues devised a new approach, utilizing a mathematical concept that wasn’t available when the game was conceived. By integrating the viscosity solution of the Hamilton-Jacobi-Isaacs equation and a loss analysis for the singular surface, they determined that an optimal solution exists for all game circumstances, resolving the dilemma.
Hamilton-Jacobi-Isaacs equation
The viscosity solution concept in partial differential equations, emerging in the 1980s, offers a fresh perspective on solving equations like the Hamilton-Jacobi-Isaacs equation. This concept is particularly relevant for optimal control and game theory problems. It involves using calculus to find derivatives of functions, which is straightforward when derivatives are well-defined, but not in cases like the wall-pursuit game.
For situations where there’s no well-defined derivative, players would typically randomly choose actions and accept the resulting losses. However, rational players seek to minimize losses. To address this issue, researchers carefully analyzed the viscosity solution around points with undefined derivatives. Subsequently, they introduced a rate of loss analysis, which in turn led to the emergence of well-defined optimal game strategies on these points.
Crucially, this analysis resolves the singular surface dilemma while staying consistent with classical analysis in relevant states. This breakthrough has broader implications for game theory. Hence, Milutinovic and his team aim to apply this to other dilemmas and encourage the research community to follow suit.
Read the original article on sciencedaily.
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