Nash equilibrium

Relying on the notion of a "fairness equilibrium," in which each player does what is fair given the behavior of the other, Rabin comments "Consider the Nash equilibrium (dare, chicken), where... 

Given that both retailers would keep adjusting their inventories in response to each other, what is the equihbrium inventory It is the point at which, given Retailer 1 s decision, the decision made by Retailer 2, fed back to Retailer 1, generates the same decision. Such an equilibrium is called a Nash Equilibrium, in honor of economist John Nash. 

Taken together, the two best response functions form a best response mapping or in the more general case RP RP, Clearly, the best response is the best player i can hope for given the decisions of other players. Naturally, an outcome in which all players choose their best responses is a candidate for the non-cooperative solution. Such an outcome is called a Nash Equilibrium (hereafter NE) of the game. 

The incentive compatibility constraints (4.29)-(4.30) state that the pair (a, a ) is a Nash equilibrium for the single-period game in which each agent receives a terminal payoff equal to his reservation utility at the end of the period. 

Straightforward MBR bidding is an ex post Nash equilibrium in this case, with each agent choosing to bid in each round for the set of bundles that maximize its payoff given the current prices. This is proved by Ausubel Milgrom [7] in their analysis of the closely-related ascending-proxy auction. 

Table 5.2. Double auction mechanisms. The traded column indicates the number of trades executed where I is the efficient number of trades. The equil column indicates whether the mechanism implements a dominant strategy or Bayesian-Nash equilibrium (BNE).

Given the algorithm terminates with the efficient allocation and gives each buyer their marginal product, it follows immediately that bidding truthfully is an ex-post Nash equilibrium of the auction. 

Loch [100] studies a Bertrand duopoly competing for homogeneous customers in a make-to-order environment. Demand arises in a Poisson fashion with an average rate that depends on price and delay. Two firms have independently distributed processing times for a unit of work, and they compete by specifying prices. The author shows that the game has a unique Nash equilibrium when firms are symmetric, that is, their processing time distributions have the same mean and variance. 

In a multiple firm setting, So [138] assumes the overall market size is constant and show that a unique Nash equilibrium solution exists and is easy to compute. Several managerial insights are offered in the paper. For example, he shows that with all other factors being equal, a firm with high capacity should compete with shorter time guarantee, while a firm with lower unit operating cost should compete with lower price. 

Cachon and Zipkin (1990), we consider games where each player chooses a stationary policy the retailer s strategy is A and the wholesaler s strategy is order-up-to quantity Q, Three situations arise either the wholesaler or the retailer might have negotiation power in the supply chain and act as Stackelberg leaders, or it is possible that the players have equal power and therefore the solution is in the form of a Nash equilibrium. We will consider all three situations. 

Proposition 3. There exists a unique Nash equilibrium pair A )... 

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