Equilibrium Solution Concepts 28

A major objective of implementing, economic based MD techniques is to mitigate the computational limitations. In fact, economics and computations, often, intertwined, in a way, that facilitates resolving mutual problems. For instance, an economic equilibrium strategy may lead to intractable computational solution approach. Similarly, an economic truth-revealing equilibrium MD, may lead to optimal computationally tractable solution. In fact, the blend of MD economic and computing techniques to developing efficient mechanisms (Conitzer & Sandholm, 2002) is a potential research space. In that vein, the game theoretic MD investigates solution concepts for private information games, and often solved by a truth revealing strategy. A mechanism , may implement a SCF in equilibrium with diverse solution concepts that predicts strategies an agent select. Each solution concept differs in assumptions about agents’ rationality and knowledge agents have about other agents. The main solution concepts may be tabled as followed:

Dominant Strategy Equilibrium(DSE): Each agent has a best-response strategy no matter

what other agent strategies S∗ _{, , ,}

∀ , ∀ , ∀ ∈ . For and strategies of player , ∀ domination is classified

as (1) strict, if , , ;(2) weak if , , and for at

least one , , , ; and (3) very weakly if , , . If

a strategy dominates all others, then it is (strongly, weakly or very weakly) dominant. The DSE provides a robust solution concept as agents don’t form beliefs about other agents’ rationality or distribution over other agent types. The single item second price auction is a DSEimplementation, as agents truthfully reveal bid values.

Nash equilibrium (NE): (Nash Jr., 1951): A strategy profile , is at NE if, for all agents , is a best response strategy to other agents, given their types and

strategies : ∗ max , , ∗ , ∀ , ∀ ∈ , ∀ .

Nash equilibrium is a stable strategy profile: no agent would want to change his strategy if she knew what strategies the other agents were following. ∀ , ∀ , ,, a strictNash strategy profile occurs if, , , and a weak Nash strategy profile occurs if , , , and is not a strict Nash equilibrium. Mixed‐strategy Nash equilibrium is necessarily always weak, while pure‐strategy Nash equilibrium can be either strict or weak, depending on the game.

Ex post Nash equilibrium:requires common knowledge about the agents’ rationality but doesn’t require any knowledge about type distributions. In this sense, ex post Nash has a no-regret property and an agent doesn’t want to deviate from its strategy even once it knows the other agents’ types. English auction is an example of ex post Nash implementation (McAfee & McMillan, 1987), with direct bidding IC strategy of ask price

whenever for value , as long as other IC agents are direct. However, direct bidding is not DSE (e.g. with jump bids). Formally, a profile of strategies … is at ex- post-Nash equilibrium if ∀ … , … are in Nash equilibrium in the

full information game. ∀ , … , ′_{: , , ) ,} ′ _{, ).}

but only knowing the forms of the other players’ strategies as functions. Let … be an ex-post-Nash equilibrium of game ∑ . . . ∑ ; Θ … Θ ; … . If

′ _{| ∈ Θ , … is DSE in game ∑}′_{. . . ∑}′ _{; Θ … Θ ; … .}

Bayesian Nash equilibrium (BNE): Agents select best-response strategies and announce types ∈ Θ to maximize their expected utility given their beliefs about the common prior about distributional information of other agent types, and assuming other agents are following expected-utility best-response maximizing strategies, announced type need

not equal true type: ∗ max , , ∗ _{, ∀ ∈ .}

BNE is the weakest solution concept adopted in MD. In a BNE, every agent must hold both beliefs about other agents’ rationality and correct beliefs about the distribution on types of other agents. An example of BNE implementation is the first price sealed-bid auction. Comparing BNE with NE, the key difference is that agent ′ strategy

must be a best response to the distribution over strategies of other agents. A refined solution concept is perfect BNE as applied to dynamic games of incomplete information (Fudenberg & Tirole, 1991). Strict incomplete information means no probabilistic information captured in the model, called also “pre-Bayesian”.

Pareto optimality: Strategy profile is Pareto optimal, strictly Pareto efficient, if there does not exist another strategy profile ∈ that Pareto dominates . A Strategy profile Pareto dominates strategy profile (not action profile) if ∀ ∈ , , and there exists some ∈ for which , means in a given Pareto‐dominated strategy profile some player can be made better without making any other player worse off. Every game must have at least one optimum.

Other Solution Concepts (Leyton-Brown & Shoham, 2008): (a) Maxmin: a strategy of player in an n‐player, general‐sum game that maximizes ’s worst‐case payoff in hostile situations where all other players play the strategies that cause the greatest harm to . The maxmin value or security level of the game for player is max min , , that minimum amount of payoff guaranteed by a maxmin strategy, while the maxmin strategy

is arg max min , ; (b) Minmax: A useful strategy when we want to consider the amount that one player can punish another without regard to his own payoff. the minmax strategy for player against player is a mixed‐strategy profile in the

arg min max , , where‐j denotes the set players other than . The minmax value for player j is min max , ; (c)Minimax Regret: In settings in which the other agent is not believed to be malicious, but is entirely unpredictable, it makes sense for agents to care about minimizing their worst‐case loss, rather than maximizing their worst‐case payoff; (d) ‐Nash; reflects the idea that players might not care about changing their strategies to a best response when the amount of utility that they could gain by doing so is very small. This leads us to the idea of ‐Nash equilibrium: Fix

0. A strategy profile is an ‐Nash equilibrium if, for all agents and for all strategies , , , ; and (e) Evolutionarily stable strategy: Roughly, a mixed strategy that resists invasion by new strategies.