The Replicator Dynamic, Chain Components and the Response Graph

In this paper we examine the relationship between the flow of the replicator dynamic, the continuum limit of Multiplicative Weights Update, and a game's response graph. We settle an open problem establishing that under the replicator, sink chain components -- a topological notion of long-run outcome of a dynamical system -- always exist and are approximated by the sink connected components of the game's response graph. More specifically, each sink chain component contains a sink connected component of the response graph, as well as all mixed strategy profiles whose support consists of pure profiles in the same connected component, a set we call the content of the connected component. As a corollary, all profiles are chain recurrent in games with strongly connected response graphs. In any two-player game sharing a response graph with a zero-sum game, the sink chain component is unique. In two-player zero-sum and potential games the sink chain components and sink connected components are in a one-to-one correspondence, and we conjecture that this holds in all games.Comment: 24 pages, 2 figure

The graph structure of two-player games

In this paper we analyse two-player games by their response graphs. The response graph has nodes which are strategy profiles, with an arc between profiles if they differ in the strategy of a single player, with the direction of the arc indicating the preferred option for that player. Response graphs, and particularly their sink strongly connected components, play an important role in modern techniques in evolutionary game theory and multi-agent learning. We show that the response graph is a simple and well-motivated model of strategic interaction which captures many non-trivial properties of a game, despite not depending on cardinal payoffs. We characterise the games which share a response graph with a zero-sum or potential game respectively, and demonstrate a duality between these sets. This allows us to understand the influence of these properties on the response graph. The response graphs of Matching Pennies and Coordination are shown to play a key role in all two-player games: every non-iteratively-dominated strategy takes part in a subgame with these graph structures. As a corollary, any game sharing a response graph with both a zero-sum game and potential game must be dominance-solvable. Finally, we demonstrate our results on some larger games.Comment: 16 pages, 11 figure

Preference Games and Sink Equilibria

In this thesis we present a new foundation for game theory. Our model for a game is defined precisely by the response graph, a natural object underlying all classical games. We call this model a preference game. Preference games generalise classical games in that all classical games have an associated preference game. Preference games also have a natural choice of solution concept: the sink equilibria, which are the sink strongly connected components of this graph. These exist in all games, and generalise pure Nash equilibria. We argue that preference games and sink equilibria form a predictive and well-founded model of strategic interaction that both clarifies problems from classical game theory and presents new solutions and predictions. Our approach is three-pronged. First, we show preference games are axiomatically well-motivated and so are broadly-applicable models of strategic interaction. We obtain these games by weakening the classical formulation of game theory with two axioms: ordinality, asserting the model requires only that players know a discrete order over the outcomes; and relevance, asserting that our model depends only on preferences over outcomes players can choose between. We give a new discussion of the Prisoner's Dilemma, where we show that the paradox can be rephrased as a consequence of Nash equilibria satisfying the relevance axiom, while Pareto efficiency satisfies only the ordinality axiom. We show further that preference games implicitly capture in their graph structure the equivalences caused by renaming players or strategies, while expressing these equivalences in classical games is cumbersome. Second, we show preference games give new insight into strategic interaction. We examine two-player games, with a focus on zero-sum and potential games. Despite both being classically defined in terms of utility functions, we show that their strategic structure can be easily understood through preference games, which make clear the duality between these two classes. We use this to prove a new theorem of classical game theory: in every two-player game, every non-iteratively-dominated strategy takes part in a 2x2 subgame with the preference structure of Matching Pennies or 2x2 Coordination. As a consequence, any two-player game sharing a response graph with both a zero-sum game and a potential game is dominance-solvable. The proofs are combinatorial. Thirdly, we show preference games are compatible with game dynamics, while classical games and mixed Nash equilibria are not. Game dynamics---the mathematical model of strategic adjustment used in evolutionary game theory---cannot generically converge to mixed Nash equilibria. By contrast, pure Nash equilibria are attracting fixed-points; the natural generalisation of this dynamic concept is the sink chain components, a topological object defined by the Fundamental Theorem of Dynamical Systems. While these do not exist in general, we prove an open problem establishing their existence under the replicator dynamic, the best-known game dynamic. We prove that sink chain components under the replicator always contain sink equilibria, and we conjecture that this relationship is always one-to-one. Thus we obtain the surprising result that the complex, sometimes chaotic, behaviour of the replicator dynamic is governed in the long run by the response graph of the game, with the outcome determined by a simple combinatorial object---sink equilibria. Sink equilibria also describe the long-run outcomes of all discrete dynamics defined by Markov chains on the response graph. We conclude with a number of open problems that emerge from preference games

Modular Decomposition of Hierarchical Finite State Machines

In this paper we develop an analogue of the graph-theoretic `modular decomposition' in automata theory. This decomposition allows us to identify hierarchical finite state machines (HFSMs) equivalent to a given finite state machine (FSM). We provide a definition of a module in an FSM, which is a collection of nodes which can be treated as a nested FSM. We identify a well-behaved subset of FSM modules called thin modules, and represent these using a linear-space directed graph we call a decomposition tree. We prove that every FSM has a unique decomposition tree which uniquely stores each thin module. We provide an $O(n^2k)$ algorithm for finding the decomposition tree of an $n$-state $k$-alphabet FSM. The decomposition tree allows us to extend FSMs to equivalent HFSMs. For thin HFSMs, which are those where each nested FSM is a thin module, we can construct an equivalent maximally-hierarchical HFSM in polynomial time.Comment: 38 pages, 11 figures. Submitted to Theoretical Computer Scienc