asymmetric games

Consider a retailer and a supplier negotiating over prices and quantities. The supplier has the information about production costs and supply chain logistics, while the retailer understands the market demand and consumer preferences. Here, the supplier might determine a wholesale price (\(P_w\)) and the quantities to be delivered, while the retailer sets a retail price (\(P_r\)). Each player aims to maximize their profit, which can be expressed as a function of these variables:\[ \text{Supplier Profit} = (P_w - C) \times Q \]where \(C\) is the cost of production and \(Q\) is the quantity supplied.\[ \text{Retailer Profit} = (P_r - P_w) \times Q \].This interplay between pricing strategies and understanding who holds more power or information illustrates an asymmetric game.

In a competitive job market, employers and job seekers are engaged in an asymmetric game. Employers decide on salaries, work conditions, and job requirements, wielding substantial power over the job offer process. Meanwhile, job seekers choose which positions to apply for based on their skills and preferences. Employers aim to minimize costs while attracting suitable candidates, whereas job seekers aim to maximize their salary and job satisfaction. The decisions made by each party can be modeled using mathematical expressions, for example:\[ \text{Employer's Objective: Minimize} \sum (S_k + B_k) \]where \(S_k\) represents salaries and \(B_k\) represents benefits given to the employee \(k\).\[ \text{Job Seeker's Objective: Maximize Utility} \]\[ U = f(S_i, J_i) \]where \(U\) stands for utility, \(S_i\) for salary, and \(J_i\) for job satisfaction.

Auctions provide another clear example of asymmetric games. Bidders have different valuations of the auctioned item and varied levels of information about other bidders' valuations. This asymmetry translates into different bidding strategies, optimizing their chance to win while not overpaying. If each bidder has a private valuation \(V_i\) for the item, strategies might involve calculating the expected payoff:\[ \text{Expected Payoff} = V_i - \text{Bid} \].The strategy can differ between an open auction, like an English auction, where knowing competitors’ strategies is possible, or a sealed-bid auction, creating a more uncertain environment.

A classic example can be found in the used car market, often known as the 'Market for Lemons'. Here, sellers typically know more about the quality of the car than potential buyers. Buyers, trying to avoid purchasing a 'lemon' (a car with defects), may undervalue cars overall. This leads to adverse selection, where good-quality cars might be kept out of the market. The formula representing a buyer's expected value could be:\[ \text{Expected Value} = P(Q) \times V_g + (1 - P(Q)) \times V_l \]Where \(P(Q)\) is the probability of getting a good car, \(V_g\) is the value of a good car, and \(V_l\) is that of a 'lemon'.

Consider an example involving two firms, Firm A and Firm B, in competition where Firm A has superior technology. Here, Firm A's costs are lower, captured by the cost functions:\[ C_A(Q) = a_A + b_A \times Q \]\[ C_B(Q) = a_B + b_B \times Q \]where \(a_A < a_B\) and \(b_A < b_B\). This leads Firm A to possibly set lower prices, thus gaining a competitive edge.