Lecture 10: Profit Maximization

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1 CS294 P29 Algorithmic Game Theory November, 2 Lecture : Profit Maximization Lecturer: Christos Papadimitriou Scribe: Di Wang Lecture given by George Pierrakos. In this lecture we will cover. Characterization of D-domain. 2. Myerson s optimal mechanism design. 3. Approximation in Bayesian mechanism design.. Characterization Consider the case where we have n bidders and item. The bidders have private values for the item v,v 2,...,v n, and the mechanism is given by the allocation and payment functions. x i :(v,...,v n ) [, ] p i :(v,...,v n ) R + We want the mechanism to satisfy the following constraints Incentive Compatibility (IC):x i (v i )v i p i (v i ) x i (b i )v i p i (b i ) for any b i. Indidual Rationality (IR): x i (v i )v i p i (v i ) i x i (probability of allocating the item ) Theorem. (Characterization) A mechanism is IC+IR iff. x i (v i ) is increasing in v i 2. p i (v i )=x i (v i )v i v i x i(z)dz -

2 -2 Lecture : Profit Maximization Proof. The only if part. Prove by picture, consider the various areas in the plot p i (v i )=A+A2,p i (b i )=A2,u i (b i )=A3+A4,u i (v i )=A3+A4+A5. IC is satisfied since u i (v i ) u i (b i ), IR is also satisfied. The if part. Using IC two times with bidder i s true value being v i and b i respectively, we can write x i (v i )v i p i (v i ) x i (b i )v i p i (b i ) x i (b i )b i p i (b i ) x i (v i )b i p i (v i ) Adding the above two equations together, we get (x i (v i ) x i (b i ))(v i b i ) Thus we know x i (v i ) is increasing with v i. u i (b i )=v i x i (b i ) p i (b i ) Take derivative of the utility with respect to the bid b i, we get u i(b i )=v i x i(b i ) p i(b i ) IC suggests that whatever bidder i s true value is, his utility should be maximized at that point, thus p i(z) =zx i(z) z If the true value of bidder i is v i,wehave p i(z)dz = zx i(z)dz p i (v i ) p i () = [zx i (z)] p i (v i )=v i x i (v i ) IR suggests p i () =, thus we get p i (v i )=v i x i (v i ) v i x i(z)dz. x i (z)dz x i (z)dz + p i () Notice when the image of x i is {, }, the above theorem states that for each bidder i there exists a threshold t i such that the bidder gets the item iff his value is at least t i, and the price he pays when he gets the item is t i..2 Myerson s mechanism (Optimal mechanism design) For simplicity we will focus on deterministic mechanisms. Denote Social Welfare (SW) = i v i x i (v) Revenue (Rev) = i p i (v)

3 Lecture : Profit Maximization -3 It s an ill-defined problem to maximize profit without any knowledge about users true utilities. Instead, we maximize expected profit assuming the value of bidder i is drawn from known distribution with density function f i, and the f i s are independent. max E[ p i ] i = p i (v)f (v )f 2 (v 2 ) f n (v n )dv v T i Example. Consider the auction with 2 bidders and item. v,v 2 are independent, uniform over [, ]. Vickrey: E[ i p i ]=E[min(v,v 2 )] = 3 Vickrey( 2 ) (Vickrey with reserve price 2 ): run a normal Vickrey auction as if there is a third bidder who bids 2, and if it turns out that the third bidder wins, the auctioneer keeps the item. E[p + p 2 v,v 2 2 ]Pr[v,v 2 2 ]= = 6 E[p + p 2 v > 2 >v 2 v 2 > 2 >v 2]Pr[v > 2 >v 2 v 2 > 2 >v 2]= 2 E[p + p 2 v,v 2 < 2 ]= 2 = 4 Adding the three cases together we have E[p + p 2 ]= 5 2 > 3. With a reserve price of 2, we can do better than Vickrey in terms of profit maximization. Consider the general situation with n bidders and item Rev(v) = i v f,...,v n f n (independent) f i (v i )( f i (v i )p i (v)dv i )dv i v i v i SW takes a similar form except p i (v) is replaced by v i x i (v i ). SW(v) = f i (v i )( f i (v i )v i x i (v i )dv i )dv i i v i v i High level idea of Myerson s mechanism: Reduce expected Rev maximization to expected SW maximization in some rtual space. Virtual valuation: φ i (v i ) Goal: Findφ i (v i )s.t.rev(v) =SW(φ(v)) It suffices for the rtual valuations to satisfy f i (v i )p i (v)dv i = f i (v i )x i (v)φ i (v i )dv i i, v i v i v i From our Characteriaztion theorem (for deterministic mechanisms), we for know each bidder there is a threshold (fixing v i ). Then the above condition becomes f i (v i ) dv i = f i (v i )φ i (v i )dv i i, v i

4 -4 Lecture : Profit Maximization Take derivative w.r.t on both sides, we get f i (v i )dv i f i ( )= f i ( )φ i ( ) Virtual valuation: φ i ( )= f i (v i )dv i f i ( ) φ i (v i )=v i F i(v i ) f i (v i ) Myerson s Auction: Input f,...,f n,b,...,b n. Compute φ (b ),...,φ n (b n ) 2. Get the allocation and price function (x, p) from VCG with bids φ (b ),...,φ n (b n ) 3. Output (x,p ) (x, φ (p)) Example 2. (Resit) n bidders, v i u[, ] independent. VCG on φ i (b i ) is just the second price auction, except we don t want to pay the winner any money. Thus we only consider bidders with rtual bids at least, i.e. v i Fi() f i(v i) v i 2. Essentially Myerson s auction in this case is Vickrey with reserve price 2. Corollary. When bidders are i.i.d according to any distribution, the optimal aunction is Vickrey (with reserve price). The reason is that φ i s are all the same, and the φ in step 2 cancels with φ in step 3. Is Myerson s auction truthful? We know that VCG is IC, and we want Myerson to be IC. What will make that work is for φ i to be increasing. φ i is indeead increasing for regular distributions. Even when φ i is not increasing, we can further reduce it to another rtual space. Ironing φ i (not increasing) ˆφ i (increasing) Some high-level intuition. Write the expected revenue with price z as Then it s easy to check R i (z) =z z f i (t)dt φ i (v i )= [ z R i(z)] z= f i (v i ) We can rewrite R i () as a function of F i (z), since Fi(z) z = f i (z), when we take the derivative w.r.t z, the chain rule will kill the f i (v i ) in the denominator. Thus φ i = R i when we write R as a function of F i (z) φ i is increasing when R i is convex, thus we can let ˆR i be the convex hull of R i, and ˆφ i = ˆR. Ironed SW Revenue. Equality holds if x i constant across ironed region.

5 Lecture : Profit Maximization -5.3 Aprroximation.3. Many items? Don t know optimal solution. For bidder, n items with unit-demand, [Chawla, Hartline, Kleinberg EC7] gives 2-approximation, and [Cai Daskalakis FOCS] gives PTAS for some distributions..3.2 Simple versus Optimal auctions [Hartline, Roughgarden EC9] What if you run Vickrey m (when the distributions of bidders values not i.i.d) m i = φ i (). Reject all bidders with value v i <m i 2. Run Vickrey on remaining bidders. Theorem 2. Vickrey m is 2-approximation if users valuations are regular.3.3 Optimal auction with correlated valuations? n bidders, (v,...,v n ) f, item [Ronen, Saberi FOCS2] gives 2 of optimal revenue for any n bidders [Papadimitriou Pierrakos STOC] Optimally for n = 2 bidders, NP-hard to get a PTAS for n Prior-freeness? Define benchmark for revenue [Hartline, Roughgarden STOC8]