Why study auctions? A lot of money at stake…<\/p>\n\n\n\n
Theory of auctions: the most successful<\/em> application of Bayesian games<\/p>\n\n\n\n Consider an auction where two bidders compete for one indivisible good<\/p>\n\n\n\n \\(v_{i}\\) – x<\/p>\n\n\n\n in X – U[0,900], Pr(X \u2264 x) = x \/ 900<\/mark><\/p>\n\n\n\n Knowing the opponent’s valuation does not<\/em> affect bidder i<\/em>‘s own valuation (1) First-price<\/mark><\/strong> auction: bidder 1 wins if \\(b_{i}\\) > \\(b_{2}\\) and pays \\(b_{1}\\) THEOREM 4.1<\/mark>. Bidding truthfully, \\(b_{1} = v_{i}\\), is the unique (weakly) dominant strategy in a second-price auction.<\/p>\n\n\n\n This implies that \\(b_{i} = v_{i}\\) satisfies<\/p>\n\n\n\n \\(u_{i}(b_{i}, b_{j}) = u_{i}(v_{i}, b_{j}) \u2265 u_{i}(b_{i}^{\\prime}, b_{j})\\) for all \\(b_{j}\\) and \\(b_{i}^{\\prime}\\).<\/p>\n\n\n\n \\(u_{i}(x, b_{i}) = \\begin{cases} v_{i}-b_{j},\\;if\\;x>b_{j}\\\\ \\frac{1}{2},\\;if\\;=b_{j}\\\\ 0,\\;if\\;x<b_{j} \\end{cases}\\)<\/mark><\/p>\n\n\n\n William Vickrey, 1996 Nobel Laureate<\/p>\n\n\n\n Figure: Case: \\(b_{j} < v_{i}\\)<\/p>\n\n\n\n To derive a BNE in first-price auctions, we take the second approach and assume<\/p>\n\n\n\n \\(b_{1}(v) = b_{2}(v) = b(v)\\) (symmetry<\/mark><\/strong>)<\/p>\n\n\n\n Consider bidder 1’s expected payoff from making a bid of x:<\/p>\n\n\n\n \\(u_{1}(x, b(v_{2})) = (v_{1} – x)Pr(b(v_{2}) < x) + 0 \\cdot Pr(b(v_{2}) > x) = Pr(v_{2} < b^{-1}(x)) = (v_{1} – x) \\cdot \\frac{b^{-1}(x)}{900}\\)<\/p>\n\n\n\n e.g., \\((12 – q_{1} – q_{2})q_{1}, q_{1} = q_{2} = q^{*}, \bso (12 – 2q^{*})q^{*}\\)<\/mark><\/p>\n\n\n\n The environment is identical because bidders 1 and 2 are playing the same game at the same time, and their valuations both come from a uniform distribution between 0 and 900.<\/mark> If we differentiate the above expression, 0 = \\(-1 \\cdot \\frac{b^{-1}(x)}{900} + (v_{1} – x) \\frac{1}{900b^{‘}(b^{-1}(x))}\\)<\/mark><\/p>\n\n\n\n \\(b(b^{-1}(x)) = x\\) \u21d2 \\(b^{‘}(b^{-1}(x)) \\cdot \\frac{\\partial b^{-1}}{\\partial x} = 1\\)<\/mark><\/p>\n\n\n\n in x = \\(b(v_{1})\\), 0 = \\(-\\frac{v_{1}}{900} + (v_{1} – b(v_{1})) \\frac{1}{900b^{‘}(v_{1})}\\)<\/mark><\/p>\n\n\n\n \\(v_{1}b^{‘}(v_{1}) = v_{1} – b(v_{1})\\) \u21d4 \\(v_{1}b^{‘}(v_{1}) + b(v_{1}) = v_{1}\\)<\/mark><\/p>\n\n\n\n \\(\\int v_{1}b^{‘}(v_{1}) + b(v_{1}) = \\int v_{1} dv_{1}\\) \u21d2 \\(v_{1}b(v_{1}) = \\frac{1}{2}v_{1}^{2} + C\\)<\/mark><\/p>\n\n\n\n \\(v_{1}b(v_{1}) = \\frac{1}{2}v_{1}^{2}\\) so, \\(b(v_{1}) = \\frac{1}{2}v_{1}\\)<\/mark><\/p>\n\n\n\n Solving the last differential equation, we obtain<\/p>\n\n\n\n \\(b(v_{1}) = \\frac{v_{1}}{2}\\)<\/p>\n\n\n\n eqbm bid<\/p>\n\n\n\n PROPOSITION 4.1<\/mark>. \\(b_{i} = \\frac{v_{i}}{2}\\) constitutes a Bayesian Nash equilibrium in the first-price auction.<\/p>\n\n\n\n In contrast with SPA, the eqbm of FPA depends on the distribution of v<\/p>\n\n\n\n SPA: \\(b^{”}(v)\\) = v 1) Used car: What price to trade the car for, buyer has an incentive to lower the price, seller has an incentive to increase the price. Observe that both bidding strategies are increasing in \\(v_{i}\\), implying that both auctions achieve allocative efficiency<\/em> (the bidder who values the most would win in eqbm) \\(\\frac{1}{2} \\cdot \\mathbb{E}[max(v_{1}, v_{2})]\\) = 300<\/p>\n\n\n\n \\(\\mathbb{E}[min(v_{1}, v_{2})]\\) = 300<\/p>\n\n\n\n This revenue equivalence<\/em> result is not a coincidence but holds in general<\/p>\n\n\n\n <\/p>\n\n\n\n\n
Auction as a Bayesian Game<\/h3>\n\n\n\n
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If the knowledge of \\(v_{j}\\) affects \\(v_{i}\\) (and correlated), then interdependent<\/em> values<\/p>\n\n\n\n\n
(2) Second-price<\/mark><\/strong> auction: bidder 1 wins if \\(b_{i}\\) > \\(b_{2}\\) and pays \\(b_{2}\\)<\/p>\n\n\n\nSecond-price (Vickrey) Auctions<\/h3>\n\n\n\n
First-price Auctions<\/h3>\n\n\n\n
Therefore, in equilibrium, we can assume that bidders 1 and 2 use exactly the same strategy.<\/mark><\/p>\n\n\n\n
FPA: \\(b^{‘}(v) = \\frac{v}{2}\\)
Figure: Eqbm Bidding Strategies<\/p>\n\n\n\n
2) Public good: If I contribute, no one else does, if others contribute, I’m not left out.
3) Auction: Efficiency<\/strong> is guaranteed, It’s done aggressively because the person with the highest valuation gets the goods<\/mark><\/p>\n\n\n\nRevenue Equivalence<\/h3>\n\n\n\n
We compare performance of the two auctions in terms of the revenue<\/p>\n\n\n\n\n
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