Abstract: The presentwork considers the following computational problem:
Given any finite game in normal form G and the corresponding
infinitely repeated game G∞, determine in polynomial time (wrt1 the representation
ofG) a profile of strategies for the players inG∞ that is an equilibrium
point wrt the limit-of-means payoff. The problem has been solved
for two players , based mainly on the implementability of the threats
for this case. Nevertheless,  demonstrated that the traditional notion of
threats is a computationally hard problem for games with at least 3 players
(see also ). Our results are the following: (i) We propose an alternative
notion of correlated threats, which is polynomial time computable
(and therefore credible). Our correlated threats are also more severe than
the traditional notion of threats, but not overwhelming for any individual
player. (ii) When for the underlying game G there is a correlated strategy
with payoff vector strictly larger than the correlated threats vector,
we efficiently compute a polynomial–size (wrt the description of G) equilibrium
point for G∞, for any constant number of players. (iii) Otherwise,
we demonstrate the construction of an equilibrium point for an arbitrary
number of players and up to 2 concurrently positive payoff coordinates in
any payoff vector of G. This completely resolves the cases of 3 players, and
provides a direction towards handling the cases of more than 3 players. It
is mentioned that our construction is not a Nash equilibrium point, because
the correlated threats we use are implemented via, not only full synchrony
(as in ), but also coordination of the other players˘ actions. But
this seems to be a fair trade-off between efficiency of the construction and
players˘ coordination, in particular because it only affects the punishments
(which are anticipated never to be used).
Abstract: We study the fundamental problem of computing an arbitrary Nash equilibrium in bimatrix games.
We start by proposing a novel characterization of the set of Nash equilibria, via a bijective map to the solution set of a (parameterized) quadratic program, whose feasible space is the (highly structured) set of correlated equilibria.
We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible. In particular, we introduce the notion of mutual (quasi-) concavity of a bimatrix game, which assures (quasi-) convexity of our quadratic program, for at least one value of the parameter. For mutually concave bimatrix games, we provide a polynomial-time computation of a Nash equilibrium, based on the polynomial tractability of convex quadratic programming. For the mutually quasiconcave games, we provide (to our knowledge) the first FPTAS for the construction of a Nash equilibrium.
Of course, for these new polynomially tractable subclasses of bimatrix games to be useful, polynomial-time certificates are also necessary that will allow us to efficiently identify them. Towards this direction, we provide various characterizations of mutual concavity, which allow us to construct such a certificate. Interestingly, these characterizations also shed light to some structural properties of the bimatrix games satisfying mutual concavity. This subclass entirely contains the most popular subclass of polynomial-time solvable bimatrix games, namely, all the constant-sum games (rank-0 games). It is though incomparable to the subclass of games with fixed rank [KT07]: Even rank-1 games may not be mutually concave (eg, Prisoner's dilemma), but on the other hand, there exist mutually concave games of arbitrary (even full) rank. Finally, we prove closeness of mutual concavity under (Nash equilibrium preserving) positive affine transformations of bimatrix games having the same scaling factor for both payoff matrices. For different scaling factors the property is not necessarily preserved.
Abstract: We study the fundamental problem 2NASH of computing a Nash equilibrium (NE) point in bimatrix games. We start by proposing a novel characterization of the NE set, via a bijective map to the solution set of a parameterized quadratic program (NEQP), whose feasible space is the highly structured set of correlated equilibria (CE). This is, to our knowledge, the first characterization of the subset of CE points that are in “1–1” correspondence with the NE set of the game, and contributes to the quite lively discussion on the relation between the spaces of CE and NE points in a bimatrix game (e.g., ,  and ).
We proceed with studying a property of bimatrix games, which we call mutually concavity (MC), that assures polynomial-time tractability of 2NASH, due to the convexity of a proper parameterized quadratic program (either NEQP, or a parameterized variant of the Mangasarian & Stone formulation ) for a particular value of the parameter. We prove various characterizations of the MC-games, which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero-sum (SZS) games of Moulin & Vial . This gives an alternative explanation of the polynomial-time tractability of 2NASH for these games, not depending on the solvability of zero-sum games. Moreover, the recognition of the MC-property for an arbitrary game is much faster than the recognition SZS-property. This, along with the comparable time-complexity of linear programs and convex quadratic programs, leads us to a much faster algorithm for 2NASH in MC-games.
We conclude our discussion with a comparison of MC-games (or, SZS-games) to kk-rank games, which are known to admit for 2NASH a FPTAS when kk is fixed , and a polynomial-time algorithm for k=1k=1 . We finally explore some closeness properties under well-known NE set preserving transformations of bimatrix games.
Abstract: In sponsored search auctions, advertisers compete for a number
of available advertisement slots of different quality. The
auctioneer decides the allocation of advertisers to slots using
bids provided by them. Since the advertisers may act
strategically and submit their bids in order to maximize their
individual objectives, such an auction naturally defines a
strategic game among the advertisers. In order to quantify
the efficiency of outcomes in generalized second price auctions,
we study the corresponding games and present new
bounds on their price of anarchy, improving the recent results
of Paes Leme and Tardos  and Lucier and Paes
Leme . For the full information setting, we prove a surprisingly
low upper bound of 1.282 on the price of anarchy
over pure Nash equilibria. Given the existing lower bounds,
this bound denotes that the number of advertisers has almost
no impact on the price of anarchy. The proof exploits
the equilibrium conditions developed in  and follows by
a detailed reasoning about the structure of equilibria and a
novel relation of the price of anarchy to the objective value
of a compact mathematical program. For more general equilibrium
classes (i.e., mixed Nash, correlated, and coarse correlated
equilibria), we present an upper bound of 2.310 on
the price of anarchy. We also consider the setting where
advertisers have incomplete information about their competitors
and prove a price of anarchy upper bound of 3.037
over Bayes-Nash equilibria. In order to obtain the last two
bounds, we adapt techniques of Lucier and Paes Leme 
and significantly extend them with new arguments