Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence

Gambling in Contests

This paper presents a strategic model of risk-taking behavior in contests. Formally, we analyze an n-player winner-take-all contest in which each player decides when to stop a privately observed Brownian Motion with drift. A player whose process reaches zero has to stop. The player with the highest stopping point wins. Contrary to the explicit cost for a higher stopping time in a war of attrition, here, higher stopping times are riskier, because players can go bankrupt. We derive a closed-form solution of the unique Nash equilibrium outcome of the game. In equilibrium, the trade-off between risk and reward causes a non-monotonicity: highest expected losses occur if the process decreases only slightly in expectation

A Polynomial-Time Randomized Reduction from Tournament Isomorphism to Tournament Asymmetry

The paper develops a new technique to extract a characteristic subset from a random source that repeatedly samples from a set of elements. Here a characteristic subset is a set that when containing an element contains all elements that have the same probability. With this technique at hand the paper looks at the special case of the tournament isomorphism problem that stands in the way towards a polynomial-time algorithm for the graph isomorphism problem. Noting that there is a reduction from the automorphism (asymmetry) problem to the isomorphism problem, a reduction in the other direction is nevertheless not known and remains a thorny open problem. Applying the new technique, we develop a randomized polynomial-time Turing-reduction from the tournament isomorphism problem to the tournament automorphism problem. This is the first such reduction for any kind of combinatorial object not known to have a polynomial-time solvable isomorphism problem

Contests with multi-tasking

The standard contest model in which participants compete in a single dimension is well understood and documented. Multi-dimension extensions are possible but are liable to increase the complexity of the contest structure, mitigating one of its main advantages: simplicity. In this paper we propose an extension in which competition ensues in several dimensions and a competitor that wins a certain number of these is awarded a prize. The amount of information needed to run the contest is hence limited to the number of dimensions won by each player. We look at the design of this contest from the point of view of maximizing effort in the contest (per dimension and totally), and show that there will be a tendency to run small contests with few dimensions. The standard Tullock model and its results are encompassed by our framework.contest design; multi-tasking; effort incentives

Contests with multi-tasking

The standard contest model in which participants compete in a single dimension is well understood and documented. Multi-dimension extensions are possible but are liable to increase the complexity of the contest structure, mitigating one of its main advantages: simplicity. In this paper we propose an extension in which competition ensues in several dimensions and a competitor that wins a certain number of these is awarded a prize. The amount of information needed to run the contest is hence limited to the number of dimensions won by each player. We look at the design of this contest from the point of view of maximizing effort in the contest (per dimension and totally), and show that there will be a tendency to run small contests with few dimensions. The standard Tullock model and its results are encompassed by our framework. -- Die Arbeit untersucht Wettbewerbsstrukturen, in denen die Teilnehmer eines Turniers in mehr als einer beobachtbaren Dimension oder Outputgröße miteinander konkurrieren, und in denen ein Wettbewerber einen Preis gewinnt, wenn er hinsichtlich einer vorgegebenen Anzahl von Dimensionen besser als seine Konkurrenten abschneidet. Es zeigt sich dabei, dass ein Turnierveranstalter, der die Anstrengungen der Turnierteilnehmer maximieren möchte, Turniere mit einer kleinen Anzahl von Outputdimensionen bevorzugen sollte.contest design,multi-tasking,effort incentives

Property rights and loss aversion in contests

We analyze the effects of property rights and the resulting loss aversion on contest outcomes. We study three situations: in “gain” two players start with no prize and make sunk bids to win a prize; in “loss” both the players start with prizes and whoever loses the contest loses their prize; and in “mixed” only one player starts with a prize that stays with him if he wins, but is transferred to the rival otherwise. Since the differences among the treatments arise only from framing, the expected utility and the standard loss aversion models predict no difference in bids across treatments. We introduce a loss aversion model in which the property rights are made salient, and as a result the reference point varies across treatments. This model predicts average bids in descending order in the loss, the mixed, and the gain treatment; and higher bids by the player with property rights in the mixed treatment. The results from a laboratory experiment broadly support these predictions. There is no significant difference in bids in the loss (gain) treatment and bids by property rights holder (nonholder) in the mixed treatment. A model incorporating both loss aversion and social preferences explains this result

Discouragement effect and intermediate prizes in multi-stage contests : evidence from Davis Cup

The discouragement effect of being the lagging player in multi-stage contests is a well-documented phenomenon. In this study, we utilize data from 2,447 Davis Cup matches in team tennis tournaments to test the effect of being behind or ahead on individuals’ performance with and without intermediate prizes. Using several different strategies to disentangle the effect of being ahead in the interim score from the effect of selection, we find the usual discouragement effect. However, the discouragement effect disappears after the introduction of intermediate prizes in the form of ranking points. The lagging favorite had close to a 20-percentage point greater probability of winning compared to matches without such a prize. We show that this result is not driven by the selection of better players into tournaments with intermediate prizes. As predicted by previous theoretical studies, our empirical findings suggest that intermediate prizes may mitigate or even eliminate the ahead-behind effects that arise in multi-stage contests. Keywords: collective decision-making, multi-stage contests, discouragement, tennispublishedVersio

Intransitivity and coexistence in four species cyclic games

Intransitivity is a property of connected, oriented graphs representing species interactions that may drive their coexistence even in the presence of competition, the standard example being the three species Rock-Paper-Scissors game. We consider here a generalization with four species, the minimum number of species allowing other interactions beyond the single loop (one predator, one prey). We show that, contrary to the mean field prediction, on a square lattice the model presents a transition, as the parameter setting the rate at which one species invades another changes, from a coexistence to a state in which one species gets extinct. Such a dependence on the invasion rates shows that the interaction graph structure alone is not enough to predict the outcome of such models. In addition, different invasion rates permit to tune the level of transitiveness, indicating that for the coexistence of all species to persist, there must be a minimum amount of intransitivity.Comment: Final, published versio