Reference no: EM132479549

Consider the following sponsored search auction instance I:

3 slots. The top slot has a known click-through rate (CTR) ctr1 = 1, the middle slot has a known click-through rate ctr2 = 0.55071 and the bottom slot has a known click-through rate ctr3 = 0.4704.

3 advertisers. Let the (private) value-per-click of advertiser 1 be 100000 (in some currency), the (private) value-per-click of advertiser 2 be 52960 and the (private) value-per-click of advertiser 3 be 14583.

The payoff of bidder i in slot j is ctrj(vi - pi), where pj is the price charged per-click in slot j. Under the Generalized Second-Price (GSP) auction rule:

Advertisers are asked to declare their value per click (this doesn't mean that their declarations are truthful!). Advertisers are then ranked according to their declarations and the advertiser with the highest declaration is assigned to the slot with the highest CTR, the advertiser with the second highest declaration is assigned to the slot with the second highest CTR, and, finally, the advertiser with the lowest declaration is assigned to the slot with the smallest CTR. For j = 1, 2, the per-click payment p3 at slot j, is set to be equal to the declaration of the advertiser assigned to slot j+ 1, while p3 = 0.

Question 1. Compute the optimal/highest social welfare in I.

Question 2. Write MATLAB code that computes a pure Nash equilibrium in I, when advertisers can declare one of the following: 0, their true value-per-click, half of their true value-per-click.

You can (or not) follow a brute-force approach, i.e. consider all possible combinations of declarations and for each of them check if it is an equilibrium. Copy and paste your MATLAB code in your report, and explicitly mention where in your MATLAB code you guarantee that the equilibrium conditions are satisfied (even if your code doesn't run or doesn't compute an equilibrium). If your code successfully computes an equilibrium, present the equilibrium that you have computed in the report and mention its social welfare. Marks will be awarded for partially-correct approaches.