Complexity of equivalence relations and preorders from computability theory

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.Comment: To appear in J. Symb. Logi

Welfare effects of strategic voting under scoring rules

Strategic voting, or manipulation, is the process by which a voter misrepresents his preferences in an attempt to elect an outcome that he considers preferable to the outcome under sincere voting. It is generally agreed that manipulation is a negative feature of elections, and much effort has been spent on gauging the vulnerability of voting rules to manipulation. However, the question of why manipulation is actually bad is less commonly asked. One way to measure the effect of manipulation on an outcome is by comparing a numeric measure of social welfare under sincere behaviour to that in the presence of a manipulator. In this paper we conduct numeric experiments to assess the effects of manipulation on social welfare under scoring rules. We find that manipulation is usually negative, and in most cases the optimum rule with a manipulator is different to the one with sincere voters

Complexity of mixed equilibria in boolean games

Boolean games are a succinct representation of strategic games wherein a player seeks to satisfy a formula of propositional logic by selecting a truth assignment to a set of propositional variables under his control. The difficulty arises because a player does not necessarily control every variable on which his formula depends, hence the satisfaction of his formula will depend on the assignments chosen by other players, and his own choice of assignment will affect the satisfaction of other players' formulae. The framework has proven popular within the multiagent community and the literature is replete with papers either studying the properties of such games, or using them to model the interaction of self-interested agents. However, almost invariably, the work to date has been restricted to the case of pure strategies. Such a focus is highly restrictive as the notion of randomised play is fundamental to the theory of strategic games – even very simple games can fail to have pure-strategy equilibria, but every finite game has at least one equilibrium in mixed strategies. To address this, the present work focuses on the complexity of algorithmic problems dealing with mixed strategies in Boolean games. The main result is that the problem of determining whether a two-player game has an equilibrium satisfying a given payoff constraint is NEXP-complete. Based on this result, we then demonstrate that a number of other decision problems, such as the uniqueness of an equilibrium or the satisfaction of a given formula in equilibrium, are either NEXP or coNEXP-complete. The proof techniques developed in the course of this are then used to show that the problem of deciding whether a given profile is in equilibrium is coNP#P-hard, and the problem of deciding whether a Boolean game has a rational-valued equilibrium is NEXP-hard, and whether a two-player Boolean game has an irrationalvalued equilibrium is NEXP-complete. Finally, we show that determining whether the value of a two-player zero-sum game exceeds a given threshold is EXP-complete.</p

Complexity of mixed equilibria in boolean games

Boolean games are a succinct representation of strategic games wherein a player seeks to satisfy a formula of propositional logic by selecting a truth assignment to a set of propositional variables under his control. The difficulty arises because a player does not necessarily control every variable on which his formula depends, hence the satisfaction of his formula will depend on the assignments chosen by other players, and his own choice of assignment will affect the satisfaction of other players' formulae. The framework has proven popular within the multiagent community and the literature is replete with papers either studying the properties of such games, or using them to model the interaction of self-interested agents. However, almost invariably, the work to date has been restricted to the case of pure strategies. Such a focus is highly restrictive as the notion of randomised play is fundamental to the theory of strategic games – even very simple games can fail to have pure-strategy equilibria, but every finite game has at least one equilibrium in mixed strategies. To address this, the present work focuses on the complexity of algorithmic problems dealing with mixed strategies in Boolean games. The main result is that the problem of determining whether a two-player game has an equilibrium satisfying a given payoff constraint is NEXP-complete. Based on this result, we then demonstrate that a number of other decision problems, such as the uniqueness of an equilibrium or the satisfaction of a given formula in equilibrium, are either NEXP or coNEXP-complete. The proof techniques developed in the course of this are then used to show that the problem of deciding whether a given profile is in equilibrium is coNP#P-hard, and the problem of deciding whether a Boolean game has a rational-valued equilibrium is NEXP-hard, and whether a two-player Boolean game has an irrationalvalued equilibrium is NEXP-complete. Finally, we show that determining whether the value of a two-player zero-sum game exceeds a given threshold is EXP-complete.</p