CS364A: Algorithmic Game Theory Lecture #9: Beyond Quasi-Linearity∗

Our discussion so far has assumed that each agent has quasi-linear utility, meaning that it acts to maximize its valuation vi(ω) for the chosen outcome ω minus the payment pi that it has to make. Thus, a bidder’s utility is a linear function of the payment made. We have placed no restrictions on payments, other than the minimal conditions that they are nonnegative and no more than the bid bi(ω) agent i made for the chosen outcome. In some important applications, payments are constrained. We first focus on budget constraints, which limit the amount of money that an agent can pay. Sometimes, there is little need to incorporate budget constraints. In a single-item auction, where we interpret the valuation of an agent as its maximum willingness-to-pay, its valuation is presumably bounded above by its budget. In other applications, especially where an agent might wind up buying a large number of items, budgets are crucial. For example, every keyword auction used in practice asks a bidder for its bid-per-click (e.g., $.25) and its daily budget (e.g., $100). Per-item values and overall budgets model well how many people made decisions in auctions with lots of items, especially when the items are identical. The simplest way to incorporate budgets into our existing utility model is to redefine the utility of player i with budget Bi for outcome ω and payment pi as