Limit groups for relatively hyperbolic groups, I: The basic tools

We begin the investigation of Gamma-limit groups, where Gamma is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of Drutu and Sapir, we adapt the results from math.GR/0404440 to this context. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate homomorphisms {h_n : G -> Gamma}, we extract an R-tree with a nontrivial isometric G-action. We then prove an analogue of Sela's shortening argument.Comment: 41 pages. The new version of this paper has been substantially rewritten. It now includes all of the results of the previous version, and also of math.GR/0408080. The exception to this is the proof of the Hopf property, which follows imediately from Theorem 5.2 of math.GR/0503045 (and does not use anything omitted from this version

One against all in the fictitious play process

There are only few "positive" results concerning multi-person games with the fictitious play property, that is, games in which every fictitious play process approaches the set of equilibria. In this paper we chararcterize classes of multi-person games with the fictitious play property. We consider an (n+1) player game {0,1,2,...,n} based on n two-person sub-games. In each of these sub-games player 0 plays against one of the other players. Player 0 is regulated, so that he must choose the same strategy in all n sub-games. we show that if all sub-games are either zero-sum ganes, weighted potential games, or games with identical payoff functions, then the fictitious play property holds for the associated game.

BEST-OF-THREE ALL-PAY AUCTIONS

We study a three-stage all-pay auction with two players in which the ?rst player to win two matches wins the best-of-three all-pay auction. The players have values of winning the contest and may have also values of losing, the latter depending on the stage in which the contest is decided. It is shown that without values of losing, if players are heterogenous (they have di¤erent values) the best-of-three all-pay auction is less competitive (the di¤erence between the players?probabilities to win is larger) as well as less productive (the players?total expected e¤ort is smaller) than the one-stage all-pay auction. If players are homogenous, however, the productivity and obviously the competitiveness of the best-of-three all-pay auction and the one-stage all-pay auction are identical. These results hold even if players have values of losing that do not depend on the stage in which the contest is decided. However, the best-of-three all-pay auction with di¤erent values of losing over the contest?s stages may be more productive than the one-stage all-pay auction.

Sequential Two-Prize Contests

We study two-stage all-pay auctions with two identical prizes. In each stage, players compete for one prize. Each player may win either one or two prizes. We analyze the equilibrium strategies where players’ marginal values for the prizes are either declining or incliningMulti-prize contests, All-pay auctions