Repeated Budgeted Second Price Ad Auction

Asaph Arnon; Yishay Mansour

doi:10.1007/s00224-013-9472-1

Theory of Computing Systems

April 2014, Volume 54, Issue 3, pp 453–478 | Cite as

Repeated Budgeted Second Price Ad Auction

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Asaph Arnon
Yishay MansourEmail author

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First Online: 07 May 2013

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Abstract

Our main goal is to abstract existing repeated sponsored search ad auction mechanisms which incorporate budgets, and study their equilibrium and dynamics. Our abstraction has multiple agents bidding repeatedly for multiple identical items (such as impressions in an ad auction). The agents are budget limited and have a value per item. We abstract this repeated interaction as a one-shot game, which we call budget auction, where agents submit a bid and a budget, and then items are sold by a sequential second price auction. Once an agent exhausts its budget it does not participate in the proceeding auctions.

Our main result shows that if agents bid conservatively (never bid above their value) then there always exists a pure Nash equilibrium. We also study simple dynamics of repeated budget auctions, showing their convergence to a Nash equilibrium for two agents and for multiple agents with identical budgets.

Keywords

Algorithmic game theory Auctions Budget Sponsored search

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Appendix: Characterization of PNE for Two Agents

In the following we give a characterization of the PNE for two agents. First, we can derive from Lemma 6 the following property of an equilibrium.

Lemma 14

If b=(b ₁,b ₂) is a PNE and b ₁≥b ₂ then b ₂≤c ₁ and b ₁≥c ₂.

In the following we discuss a few cases, based on the relationship between c ₁ and c ₂, and the budgets versus the values.

Lemma 15

Assume that we have two agents with c ₂<c ₁ and B ₂/v ₁<N. Then any bids (b ₁,b ₂) and \((\hat{B}_{1},\hat{B}_{2})\) such that c ₁≥b ₁≥b ₂≥c ₂ and b ₂=min{v ₂,b ₁}, are a PNE, and in this case those are the only PNEs where agents submit their true budget and bid at most their value.

Proof

First, note that since B ₂/v ₁<N we have that u ₁>0, since if agent 1 deviates and bids \(b'_{1}=v_{1}-\epsilon\) he is guaranteed to be allocated items, i.e., \(x'_{1}>0\).

We show that there is no PNE with b ₁<b ₂. For contradiction, assume there is such a PNE. By Lemma 6, for this to be a PNE, we need that b ₂≥c ₁ and c ₂≥b ₁. Since the lemma assumes that c ₁>c ₂, we have b ₂≥c ₁>c ₂≥b ₁. Since agent 1 has positive utility, u ₁>0, he can deviate to \(b'_{1}=c_{1}\), strictly increasing his allocation \(x'_{1} > x_{1}\), paying p _min, and therefore increasing his utility, \(u'_{1}>u_{1}\). This contradicts that agent 1 is best responding. We conclude that b ₁≥b ₂.

Since b ₁≥b ₂, by Lemma 6, for this to be a best response we need that c ₁≥b ₂ and b ₁≥c ₂. Therefore, c ₁≥b ₁≥b ₂≥c ₂. we consider two cases based on the magnitude of B ₁/v ₂: (i) If B ₁/v ₂≥N, it implies that c ₂=v ₂ (since the second agent, when bottom rank, has zero utility). Since b ₂≥c ₂ and v ₂≥b ₂, then b ₂=c ₂=v ₂. (ii) If B ₁/v ₂<N then the second agent has positive utility, u ₂>0, since the deviation \(b'_{2}=v_{2}-\epsilon\) would guarantee a positive utility. Therefore, agent 2 would maximize his utility by maximizing his bid while maintaining his bottom rank, i.e., b ₂=min{v ₂,b ₁}. □

Lemma 16

Assume that we have two agents with c ₂<c ₁ and B ₂/v ₁≥N. Then:

(1)

if v ₁>v ₂ we have v ₁=c ₁≥v ₂≥c ₂ and b ₁≥b ₂ and b ₂∈[c ₂,v ₂] and b ₁∈[v ₁,c ₂].
(2)
If v ₂≥v ₁ we have v ₂≥v ₁=c ₁>c ₂ and two possible outcomes:
1. (2a)
  
  b ₂≥b ₁ and b ₁∈[p _min,c ₂] and b ₂∈(v ₁,v ₂], and
2. (2b)
  
  b ₁≥b ₂ and b ₁∈[c ₂,c ₁] and b ₂∈[p _min,c ₂].

In this case those are the only PNEs where agents submit their true budget and bid at most their value.

Proof

If B ₂/v ₁≥N then agent 1 critical bid is c ₁=v ₁, since for any z<v ₁, we have B ₂/z>N, and hence x ₁=0 in the bid vector (z,z) when agent 1 is bottom rank.

For case (1), we have that v ₁>v ₂. This implies that c ₁=v ₁>v ₂≥b ₂, and by Lemma 6 agent 1 prefers the top rank. Therefore b ₁≥b ₂. In order for agent 1 best response to be top rank we need that b ₂∈[c ₂,v ₂], since v ₂<v ₁=c ₁. In order for agent 2 best response to be bottom rank we need that b ₁∈[c ₂,v ₁].

For case (2), we have that v ₁≤v ₂. In (2a) we have that agent 1 allocation is x ₁=0 and hence his utility is u ₁=0. Agent 2 prefers the top rank since b ₁≤c ₂. Since b ₂>v ₁ agent 1 prefers the bottom rank. In (2b) agent 1 is top rank and may have positive utility. Agent 1, if he deviates to \(b_{1}'=b_{2}-\epsilon\) will have zero utility. Agent 2 prefers the bottom rank since c ₂≤b ₁. □

Lemma 17

Assume that we have two agents with c ₂=c ₁=c then b=(b ₁,b ₂)is a PNE if and only if either: (i) b ₁∈[c,v ₁] and b ₂∈[p _min,c], or (ii) b ₂∈[c,v ₂] and b ₁∈[p _min,c].

Combining Lemmas 15, 16 and 17 gives a characterization of the PNE for two agents.

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Asaph Arnon
- 1
- 2
Yishay Mansour
- 1
Email author

1.Tel Aviv UniversityTel AvivIsrael
2.Google Tel AvivTel AvivIsrael

Repeated Budgeted Second Price Ad Auction

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