Lecture 17 computational advertising
Def: Competitive Ratio:
- In the worst assignment, how much money you would get with a particular algorithm compared to how much money you would get with the optimal assignment.
Google AdWords Problem: Maximize Expected Revenue = CTR * bid amount
Algorithms for Ad assignment (Online Graph Assignment)
Greedy Algorithm:
- Has competitive Ratio of 1/2 on two advertisers.
|M_{opt}| \le |M_{greedy}| + |G|
|M_{greedy}| \ge |B|
$|G| \le |B|$
$|G| \le |B| \le |M_{greedy}|
BALANCE Algorithm:(1)
- Consider query $q$, bidder $i$.
- Let the bid of $i$ = $x_i$
- budget of i = $b_i$
- Amount spent so fat = $m_i$
- Fraction of budget left over is $f_i = 1-m_i/b_i$
- Define $\psi_i(q) = x_i(1-e^{-f_i})$
- Allocate query $q$ to bidder $i$ with largest value of $\psi_i(q)$
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Equal budget Heuristic: For each query, pick the advertiser with the largest unspent budget
- Has Competitive Ratio of 3/4 on two advertisers, or $1-1/e \approx 0.63$ with many advertisers. In addition, this is the best possivle competitive ratio for any online algorithm.
(1) Proof: Consider two advertisers, and optimial solution exhausts both advertiser’s budgets. Optimal Revenue= 2B BALANCE must exhaust at least one budget. Balance Revenue = 2B -x = B+y Now, since $y \ge x$ by balance algo, balance revenue is min for $y = x \rightarrow y \ge B/2$