# Value of Learning in Sponsored Search Auctions

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6 to the same expected bid per impression. For example, in the first step (i.e., when we are in state (k, N) in scenario 1), the corresponding bid in scenario 2 is either bˆλ k,n /ˆλ k+1,n+1 or bˆλ k,n /ˆλ k,n+1, depending on whether the state is (k + 1, N + 1) or (k, N + 1). The expected utility of the advertiser in scenario 2 in this step can be written as ˆλ k,n g(ˆλ k+1,n+1, bˆλ k,n /ˆλ k+1,n+1 ) + (1 ˆλ k,n )g(ˆλ k,n+1, bˆλ k,n /ˆλ k+1,n+1 ) Using (1), this can be written as: ˆλk,N b ˆλ k,n (vˆλ k+1,n+1 p)f(p)dp + (1 ˆλ k,n ) = 0 ˆλk,N b 0 ˆλk,N b ( v(ˆλk,n ˆλk+1,N+1 + (1 ˆλ k,n )ˆλ k,n+1 ) p ) f(p)dp 0 (vˆλ k,n+1 p)f(p)dp Using the definition ˆλ k,n as the posterior probability of getting a click conditioned on having had k clicks out of the first N impressions, it is easy to show that ˆλ k,n ˆλk+1,N+1 + (1 ˆλ k,n )ˆλ k,n+1 = ˆλ k,n. (6) Therefore, the expected utility in the first step in scenario 2 is equal to ˆλk,N b 0 (vˆλ k,n p)f(p)dp = g(ˆλ k,n, b), which is the same as the expected utility in the first step in scenario 1. Similarly, in any step the simulated strategy in scenario 2 obtains the same expected payoff as in scenario 1. Thus, (k, N) 0. The above theorem only shows that the optimal bid of the advertiser is never smaller than her true value. To show that this bid is sometimes strictly larger than the value, we focus on the case of uniform distributions: We assume a uniform prior D λ = U[0, 1] on the clickability and a uniform price distribution D p = U[0, 1]. Straightforward calculations using the Bayes rule and the prior D λ shows that the posterior probability density for the clickability λ in a state (k, N) is ( ) n (n + 1) λ k (1 λ) N k. k The expected value of λ given this posterior is ˆλ k,n = k+1 N+2, and the function g(.) from (1) can be written as g(ˆλ k,n, b) = ˆλ 2 k,n b(v b 2 ). Theorem 2. In the above model of repeated auction, the optimal bid of the advertiser in every state (k, N) is strictly larger than v. More specifically, we have (k, N) = Ω(N 2 ).

7 Proof (Proof Sketch). As in the proof of Theorem 1, we need to bound the difference between the optimal expected utility of scenarios 1 and 2. Again, we do this by taking the optimal strategy in scenario 1, and simulating it in scenario 2. Unlike the proof of Theorem 1, we simulate a strategy that submits a bid of b in scenario 1 by submitting the same bid in scenario 2. First, notice that with this strategy, the probability of winning the first auction in scenario 2 can be written as ˆλ k,n F (bˆλ k+1,n+1 )+(1 ˆλ k,n )F (bˆλ k,n+1 ) = (ˆλ k,n ˆλk+1,N+1 +(1 ˆλ k,n )ˆλ k,n+1 )b Using (6), the above probability is equal to ˆλ k,n b, which is the same as the probability of winning in scenario 1. This ensures that the simulated strategy in scenario 2 has the same branching probabilities as the optimal strategy in scenario 1. Next, we need to bound the difference between the expected utility of one auction in the two scenarios. Here we only do this for the first auction. The inequality for the other auctions can be proved similarly. The difference between the expected utilities of the advertiser in the first auction in the two scenarios can be written as: ˆλ k,n g(ˆλ k+1,n+1, b) + (1 ˆλ k,n )g(ˆλ k,n+1, b) g(ˆλ k,n, b) ) = b(v b 2 ) (ˆλ k,n ˆλ2 k+1,n+1 + (1 ˆλ k,n )ˆλ 2 k,n+1 ˆλ 2 k,n ( ) = b(v b ( 2 ) k + 1 )( k + 2 ) 2 ( k + 1 )( k + 1 ) 2 ( k + 1 ) N + 2 N + 3 N + 2 N + 3 N + 2 = b(v b + 1)(N k + 1) )(k 2 (N + 2) 2 (N + 3) 2 = Ω(N 2 ). 4 Value of learning Given the result in the previous section, we can define the value of learning of an advertiser as the difference between the optimal bid of the advertiser and her value-per-click. More formally, the value of learning is the difference between the Gittins index in the Markov Decision Process (MDP) defined based on the auction. If we could compute these indices, we could simply design an alternative auction mechanism that allocates according to these indices, thereby achieving the optimal MDP solution and eliminating the incentive to overbid. Unfortunately, Gittins indices are quite hard to compute. As a practical alternative, we can use proxies for the value of learning that are easy to compute. Perhaps the simplest method for doing this is to take the value of learning of an advertiser to be proportional to the variance of our estimate of the clickability of this advertiser. This has the advantage that it can be easily computed, and gives a boost to ads that we currently do not have an accurate estimate of its clickability.

11 We simulated the auction and click behavior for a range of C to illustrate the effect of imposing different degree of learning in the mechanism. The case C = 0 corresponds to the case when there is no value of learning included in the auction. The higher C is, the more impact value of learning has on price and ranking, and hence revenue and efficiency. Figure 1 (a) shows the moving average of the total revenue generated over the duration of the simulation, and figure 1 (b) shows the moving average of the total efficiency. The moving average window used in the graphs has width 400. As can be seen in the figures, the revenue is consistently higher when value of learning is used in the auction. Furthermore, efficiency is higher in the transient when the appropriate value of learning (C = 2) is used. It should be noticed that when C is large (C = 6), both the transient and final efficiency can suffer as too much exploration is being done. We also simulated the case when not all ads in each sample search are shown in every auction by reducing the number of slots that can be shown in each auction m to five (in Yahoo! search this can be as high as twelve). In other words, when the number of ads available is more than five, the algorithm is forced to select only five ads to show, with the rest not getting any exposure at all. As can be seen in Figure 2, the power of incorporating value of learning in the auction is more evident in this case. The efficiency of the auction with C other than zero is much higher than when C is zero. This is because the set of ads that are shown are fixed very early as other ads with high clickability are never given a chance to prove themselves. The revenue is also higher when C is non-zero, due to the price effect of value of learning as well as improved efficiency. (a) revenue (b) efficiency Fig. 1. Moving average of revenue and efficiency for different setting of C. References 1. D. Agarwal, B. Chen, and P. Elango. Explore/exploit schemes for web content optimization. In ICDM, 2009.

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