Continuous fictitious play in zero-sum games

Robinson (1951) showed that the learning process of Discrete Fictitious Play converges from any initial condition to the set of Nash equilibria in two-player zero-sum games. In several earlier works, Brown (1949, 1951) makes some heuristic arguments for a similar convergence result for the case of Continuous Fictitious Play (CFP). The standard reference for a formal proof is Harris (1998); his argument requires several technical lemmas, and moreover, involves the advanced machinery of Lyapunov functions. In this note we present a simple alternative proof. In particular, we show that Brown''s convergence result follows easily from a result obtained by Monderer et al. (1997).mathematical economics;

A non-cooperative foundation for the continuous Raiffa solution

This paper provides a non-cooperative foundation for (asymmetric generalizations of) the continuous Raiffa solution. Specifically, we consider a continuous-time variation of the classic Ståhl–Rubinstein bargaining model, in which there is a finite deadline that ends the negotiations, and in which each player’s opportunity to make proposals is governed by a player-specific Poisson process, in that the rejecter of a proposal becomes proposer at the first next arrival of her process. Under the assumption that future payoffs are not discounted, it is shown that the expected payoffs players realize in subgame perfect equilibrium converge to the continuous Raiffa solution outcome as the deadline tends to infinity. The weights reflecting the asymmetries among the players correspond to the Poisson arrival rates of their respective proposal processes

The Asymmetric Leximin Solution

In this article we define and characterize a class of asymmetric leximin solutions, that contains both the symmetric leximin solution of Imai[5] and the two-person asymmetric Kalai-Smorodinsky solution of Dubra [3] as special cases. Solutions in this class combine three attractive features: they are defined on the entire domain of convex n-person bargaining problems, they generally yield Pareto efficient solution outcomes, and asymmetries among bargainers are captured by a single parameter vector. The characterization is based on a strengthening of Dubra’s [3] property Restricted Independence of Irrelevant Alternatives (RIIA). RIIA imposes Nash’s IIA (Nash [9]), under the added condition that the contraction of the feasible set preserves the mutual proportions of players’ utopia values. Our axiom, entitled RIIA for Independent Players (RIP), says RIIA holds for a group of players, given that the contraction of the feasible set does not affect players outside that group

The Kalai-Smorodinsky Solution with Loss Aversion

We consider bargaining games under the assumption that bargainers are loss averse, i.e. experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on a solution. Given a bargaining game, we say outcome z is self-supporting under a given bargaining solution, whenever transforming the game using outcome z as reference point, yields a transformed game in which the solution is z.We show that n-player bargaining games have a unique self-supporting outcome under the Kalai-Smorodinsky (KS) solution. We define a bargaining solution, giving exactly this outcome, and characterize it by the standard axioms of Scale Invariance [SI], Individual Monotonicity [IM], and Strong Individual Rationality [SIR], and a novel axiom called Proportional Concession Invariance [PCI].A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome, does not change this outcome.microeconomics ;

Alternating offers bargaining with loss aversion

The Rubinstein alternating offers bargaining game is reconsidered under the assumption that each player is loss averse and the associated reference point is equal to the highest turned down offer of the opponent in the past. This makes the payoffs and therefore potential equilibrium strategies dependent on the history of play. A subgame perfect equilibrium is constructed, in which the strategies depend on the history of play throughthe current reference points. It is shown that this equilibrium is unique under some assumptions that it shares with the equilibrium in the classical model: immediate acceptance of equilibrium offers, indifference between acceptance and rejection of such offers, and strategies depending only on the current reference points. It is also shown that in this equilibrium loss aversion is a disadvantage. Moreover, a relation with asymmetric Nashbargaining is established, where a player’s bargaining power is negatively related to own loss aversion and positively to the opponent’s loss aversion.mathematical economics;

Feasible sets, comparative risk aversion and comparative uncertainty aversion in bargaining

We study feasible sets of the bargaining problem under two different as- sumptions: the players are subjective expected utility maximizers or the players are Choquet expected utility maximizers. For the latter case, we consider the effects on bargaining solutions when players become more risk averse and when they become more uncertainty averse

Az erős,a gyenge, meg a ravasz: Hatalom és stratégiai viselkedés szavazási játékokban = The Strong, the Weak and the Cunning: Power and Strategy in Voting Games

Kutatási eredményeink három téma köré csoportosíthatók. Ezek közül az első a stratégiai megfontolások vizsgálata. Megmutatjuk, hogy a szavazók növelhetik befolyásukat, ha veszekednek más szavazókkal és a stratégiai hatalmi indexek jól definiáltak a játékok bizonyos osztályaira. Egy másik vonal a kooperatív játékok olyan kulcsfontosságú tulajdonságait vizsgálja, mint a konvexitás, vagy az egzaktság. Bizonyos esetekben a nyerő koalíciók halmaza külső okok miatt korlátozott: erre a leggyakoribb példa, mikor egy hálózaton elhelyezkedő csúcsok helyzeti befolyását vizsgáljuk. A csúcsok csak az őket összekötő élek mentén kommunikálhatnak és csak szomszédaikkal. Több érték és index is kiterjesztésre, illetve bevezetésre kerül ilyen hálózati játékokra, illetve az értékekhez axiomatikus karakterizációt adunk. Végül a hatalmi indexeket olyan játékokra is kiterjesztjük, ahol egyes szavazók hiányozhatnak. A nem stratégiai hiányzást vizsgáljuk és a Shapley értéket teljesen karakterizáljuk az általánosított súlyozott szavazási játékok osztályán. Modellünket különböző parlamentekre alkalmazzuk, illetve az elméleti módszerek több egyéb alkalmazását is vizsgáltuk, úgymint a Lisszaboni Szerződés hatását a Miniszterek Tanácsában folyó súlyozott szavazás hatalmi viszonyaira. | The results of the project centre around three themes. The first is strategic considerations. We have shown that voters are able to increase their power by strategic quarrelling and the strategic power indices are well defined for certain classes of games. Additional papers provide tests on key properties, such as convexity and exactness of cooperative games. In some situations the set of feasible (winning) coalitions is restricted exogenously. The most common example is to study positional power over a network where the voters are located at the nodes and can only communicate with their neighbours. Several values and indices are introduced and characterised for games over networks. At last we generalised power indices to weighted voting games where representatives may be absent. We study non-strategic absenteeism and characterise the Shapley value for the class of generalised weighted voting games. We have also studied applications studying the effect of absent voters in various national parliaments or the effect of the Lisbon Treaty of the European Union to the power balance in the Council of Ministers

Bargaining, conditional consistency, and weighted lexicographic Kalai-Smorodinsky Solutions

We reconsider the class of weighted Kalai-Smorodinsky solutions of Dubra (Econ Lett 73:131–136, 2001), and using methods of Imai (Econometrica 51:389–401, 1983), extend their characterization to the domain of multilateral bargaining problems. Aside from standard axioms in the literature, this result involves a new property that weakens the axiom Bilateral Consistency (Lensberg, J Econ Theory 45:330–341, 1988), by making the notion of consistency dependent on how ideal values in a reduced problem change relative to the original problem

A Bargaining Game with Proposers in the Hot Seat

This note reconsiders the Rubinstein bargaining game under the assumption that a rejected offer is only costly to the proposer who made the rejected offer. It is shown that then, the classic result of Shaked that, in the multilateral version of this game, every division of the good can be sustained in SPE no longer holds. Specifically, there are many SPE, but players’ (expected) payoffs in SPE are unique. The assumption further leads to a responder advantage