Online Appendi for Anglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam Christiaan van Bochove, Lars Boerner, Daniel Quint January, 06 The first part of Theorem states that for any, ρ > 0, and > 0, an equilibrium eists. The proof is constructive: for 4, equilibrium strategies are given in the appendi of the paper. Here, we show that these strategies do indeed make up an equilibrium, and deal separately with the cases 3 and. First Round Play First-round strategies and beliefs for player are Do not drop out at P 0 At any history where P > 0 and there is one other active bidder, believe that he has signal s i, and If P < P, remain active If P P, drop out and for player i are Drop out at P 0 At any history where P > 0 and bidder i is still active along with bidder, If P P, believe s U0, ; remain active if s i and drop out if s i < If P < P < P, believe s ; remain active if s i and drop out if s i < If P P, drop out immediately where and c + P c + + + ρ if ρ + ρ if ρ < c + + P + ρ if ρ c + + ρ if ρ < Since all bidders other than bidder are epected to drop out at P 0, any history where P > 0 and more than one of bidders, 3,..., are still active could only be reached following
simultaneous deviations by more than one player. Strategies at such histories therefore does not affect equilibrium play, and we do not specify what strategies are played following such deviations. et, we will describe second-round beliefs and strategies, both on and off the equilibrium path, and show that these are equilibrium strategies. After that, we will show that these firstround strategies are best-responses given that continuation play. Second Round Play On the Equilibrium Path Bidder won the first round at a price of 0. Common beliefs are that s i U0, for all i. Second-round bidding functions are bs i c + s i + s i To see that these bid functions are an equilibrium, first note that bids above b are strictly dominated by bidding b, since this will still win with probability ; and bids below b0 earn epected payoff of 0. For 0,, a bidder with signal s i who plans to bid b will win whenever s j < for every j i, which occurs with probability ; in that event, the security is worth, on average, c + si +, so his epected payoff is c + si + c + + s i Differentiating with respect to gives si s i, which is positive when < s i and negative when > s i, so epected payoff is maimized by setting s i. ote also plugging this into the payoff function that this gives nonnegative epected payoff, so bidding less than b0 and losing for sure is not a profitable deviation. Should bidder choose to drop out at P 0 along with the other bidders so that the first-round winner is chosen by a random tiebreaker, second-round beliefs and strategies are unchanged. Second Round Play After a First-Round Deviation By Bidder i We refer to this continuation game as CG. As in the tet, assume without loss of generality that bidder is the one who deviated. Since bidder s equilibrium strategy following such a deviation is to bid up to price P, assume that either the deviating bidder has dropped out at a price P 0, P or has won the first round at the price P. All bidders commonly believe that s ; beliefs about
the other bidders signals including bidder match the prior s i U0,. Strategies are: Bidder does not bid regardless of his actual type If P c + +, then nobody bids If P c +, c + +, then let s solve P c + + s. Bidder i does not bid when s i s ; when s i > s, he bids bs i c + If P c +, then bidders i each bid s i + s i s s i + s i + bs i c + s i + s i + In the first case, P c + +, each bidder i epects nobody to bid; winning the object would require bidding strictly more than P, therefore strictly more than c + +, for an object that will be worth, on average, c + si + + c + +, so not bidding is a best-response. Bidder epects the object to be worth c + s + < c + +, so not bidding is a best-response for him as well. In the second case, bs i is strictly increasing, so bids above b are again strictly dominated. By construction, bs P. For s,, a bidder with signal s i who bids b epects to win when the other bidders ecluding himself and bidder have signals below ; he therefore earns epected payoff c + si + + c + s i s s + + s i s Differentiating with respect to gives 3 s i, which is positive for < s i and negative for > s i, so epected payoff is maimized at bs i. Bidding P bs or a tiny bit above that wins only if nobody else bids, giving epected payoff s c + s si + + s s i s c s s + s s s + s + s s i s making it worse than not bidding when s i < s. So for s i < s, any bid equal or greater than P would earn negative epected profits; and for s i > s, bidding bs i earns greater payoff than bs, which is itself positive, so bs i is a best-response. 3
ote that if bidder dropped out at a price P < P but above c +, then bidder is the first-round winner. The fact that he will win the object if nobody bids in the second round means that by not bidding, he pays bs for the object whenever all other bidders aside from have signals s i < s. This is equivalent to bidder having to bid bs if he does not bid higher than that; his optimal move is therefore still to bid bs when s > s, and to not bid in the second round when s s. As for bidder, bidding b would give epected payoff c + so not bidding is a best-response. s + c s + + s s + + < 0 In the third case, bids above b are again dominated, and a bid of b for 0, earns epected payoff c + si + + c s i + + The derivative with respect to is again n 3 s i, so epected payoff is again maimized at s i, which gives positive epected payoff so bidding less than b0 is not a profitable deviation. For bidder, bidding b would give c + s + c + + s + < 0 so not bidding is again a best-response. Bidder s Payoff From Deviating In the First Round By deviating in the first round and dropping out before P, bidder guarantees entry into CG discussed above, in which he does not bid and earns epected payoff 0; so remaining active past price 0, but planning to drop out before P, is not a profitable deviation. On the other hand, if bidder deviates in the first round and does not drop out before P, he epects bidder to drop out at price P. We net show that this would earn epected profit of 0 or less for bidder. 4
If ρ, then P c + + + ρ c + +, so if the first round reaches price P, nobody bids in the second round. In that case, by winning the first round at price P and not updating beliefs about s, bidder earns second-round payoff c + s + c + + + ρ s ρ Combined with winning the premium, then, bidder s epected payoff would be nonpositive. If ρ <, then if P P c + + ρ, s solves and so s c + + ρ c + + s ρ. After winning the first round at price P, bidder epects to be stuck with the security with probability s, for second-round epected payoff s c + s + s c + + s s s s s s s s s ρ s s ρ So including winning the premium in the first round, winning the first round at price P gives epected payoff 0 to bidder if s, and negative epected payoff otherwise. ote therefore that if s and bidder has already deviated by failing to drop out immediately, he epects payoff 0 whether he drops out before P or waits for bidder drop out at P. This means that once he has already deviated, it is a weak best response not to drop out before P which makes bidder willing to punish him by staying in until P ; but also that deviating in the first place cannot be profitable. Second Round Play After Deviations By Bidders i and Again, assume i is the first deviating bidder. If bidder does not drop out immediately, and bidder either drops out at a price P < P, or fails to drop out at price P so that either bidder or wins the first round at a price P > P, we refer to the resulting continuation game as CG3. Beliefs and strategies in the second round are as follows: Everyone believes s s Bidders and regardless of their actual types do not bid 5
If P c + + 3 +, then nobody bids If P c +, c + + 3 +, then let s solve P c + s + 3 s + Bidder i 3 does not bid when s i s ; when s i > s, he bids bs i c + If P c +, then bidders i 3 each bid 3 s i + s i s s i + 3 s i + bs i c + 3 s i + 3 s i + In the first case, if P c + 3 + +, then a bidder i 3 with signal s i epects no one else to bid, and believes that the security is worth in epectation c + s i + 3 + + P, and therefore does not bid. Bidder i {, } believes the security to be worth in epectation c + s i + + < P and therefore does not bid either. For the second case, a bid above c + + 3 + is dominated; a bid by bidder i 3 equal to b > s wins whenever s j < for all j {3, 4,..., } {i}, which is with probability 3, and therefore delivers epected payoff of c 3 + si + 3 + c + 3 + 3 s i 3 s 3 Differentiating with respect to gives s + 3 + s i 3 3 s 3si 3 3 which is positive for < s i and negative for > s i, so bs i is a best-response among bids above P. Bidding P or slightly above P would give epected payoff s i s 3 3 s s si s 3 s which is positive for s i > s and negative for s i < s. So not bidding is a best-response for s i s, and bs i a best-response for s i > s. Bidder i {, } who is not the first-round winner would, by 6
bidding b, earn epected payoff c + si + + c + 3 + s i + 3 s s + 3 + 0 and so prefers not to bid. Bidder i {, } who is the first-round winner is effectively bidding bs by not making a new bid; but since { d d s i + 3 } s 3 s i + 3 s < 0 he prefers bs to a higher bid, and does not bid again. For the third case, a bid of b by bidder i 3 gives epected payoff 3 c + si + 3 + c 3 + 3 + s 3 i 3 which has derivative 3 s i 3 3, which once again is positive for < s i and negative for > s i ; setting s i gives strictly positive payoff, so bs i is a best-response. For bidder i {, } whether or not the first-round winner, since the first-round winner will for sure be outbid if he does not bid again, but counting the premium received as a sunk benefit, bidding b gives payoff c + si + + c s i + 3 so not bidding is a best-response. 3 + 3 + 0 Bidder s Epected Payoff From Failing To Punish et, we show that if s s and either bidder or wins the first round at price P, his epected payoff including the first-round premium is 0. This ensures that i once the price passes P and bidder believes s, then assuming s as well, bidder is willing to stay in until P ; and ii bidder cannot gain by failing to punish a deviation by bidder, since he epects bidder to stay in until P, and even if s s, winning the first round would therefore give bidder zero epected payoff. If ρ, then P c + +3 3, so if P P, nobody will bid in the second round; in 7
that case, the first-round winner wins the security for sure regardless of others signals, so if s s, his epected payoff is c + + c + + + ρ ρ which, combined with winning the premium ρ in the first round, gives a net epected payoff of 0. If ρ <, then when P P, s solves c + + ρ c + s + 3 s + and therefore s ρ. So with probability s, nobody bids in the second round, and the security is worth, in epectation assuming s s, c + s +. So the first-round winner i {, } at price P, assuming s s, gets epected payoff gross of winning the premium s c + s + c s + s s s ρ ρ and so including winning the premium, the first-round winner would get e-ante epected payoff 0. First Round Strategies Are Equilibrium Strategies Finally, we need to show the strategies given earlier for the first round are indeed best-responses. First, consider bidder at P 0. If he drops out immediately leading to the first round being determined by a random tiebreaker, he wins the premium with probability instead of, and does not affect second-round play, so this is not a profitable deviation. Following a deviation by bidder, bidder believes s, and epects bidder to stay active until the price reaches P. Dropping out at price P would lead to CG and nonnegative payoffs. Dropping out before or after P would lead to CG3 and zero payoffs; staying in until P would lead to zero payoffs or negative payoffs if s <. So sticking to equilibrium strategies is a best-response. As for bidder representing all bidders i >, dropping out immediately leads to nonnegative strictly positive if s > 0 epected payoffs in the second round. ot dropping out immediately, bidder would epect bidder to remain active until P and then drop out; dropping out before 8
then would lead to CG and zero payoffs; winning the first round at price P would likewise lead to CG and zero payoffs or negative if s <. So having already failed to drop out at P 0, it remains a best-response to stay active until P P. If bidder fails to drop out at P P, bidder then believes s ; provided s as well, bidder epects zero payoffs if he stays in and wins at price P, strictly positive payoffs if he stays in and wins at a lower price than that; dropping out prior to P would lead to CG3, and zero payoffs for bidder. So having stayed in past P 0, bidder has no reason to drop out prior to P. Once the price passes P, the winner of the first round is assured negative payoffs, so both players best-respond by dropping out immediately. As mentioned above, any history not eplicitly addressed in the strategies above could only be reached via simultaneous deviations by multiple bidders, and therefore cannot affect play on the equilibrium path; we therefore do not specify what happens at these histories. What if 3? If 3, define { P ma P, c + 3 + } 6ρ If P P, modify both bidder s strategy following a deviation by bidder to be, stay in till P, drop out at P, and drop out immediately at any P > P, and bidder s strategy following a deviation by himself to be, stay in till P, do not drop out at P, drop out immediately at any P > P. Play at all histories other than CG3 are otherwise unchanged. In CG3 when 3, bidders effectively play the drainage tract auction equilibrium considered in Engelbrecht-Wiggans, Milgrom and Weber 983, Milgrom and Weber 98b, and Hendricks, Porter and Wilson 994, where the one bidder with private information plays a pure strategy and the uninformed bidders play mied strategies and earn zero epected profit. We will construct such an equilibrium where the round-one winner plays a mied strategy while the third bidder whiever of bidders and lost the first round does not bid. Let i denote the identity of the first-round winner, and j whichever of and is not i. Recall that this continuation game includes the common beliefs that s s. Let s 3 solve P c+ 3 +s 3, or s 3 0 if P < c + 3. Define a function Γ : s 3, R + by Γt c + 3 + t + s 3 t 9
and note that Γ t 3 Γs 3 P. s 3 t > 0 for t s 3,, so Γ is strictly increasing. Also note that Define a probability distribution G with support P, Γ by if Γ G ep Γ dt if P, Γ c+ 3 +Γ t t 0 if < P ote that this distribution has a point mass at P. Bidding strategies are as follows: Bidder 3 bids Γs 3 if s 3 > s 3, and does not bid if s 3 s 3 If s i, bidder i mies such that the CDF of his bids is G with the point mass at P indicating no new bid; if s i <, bidder i does not bid Bidder j does not bid To see this is an equilibrium, we first show that if bidder 3 plays this strategy, bidder i is indifferent between not bidding and bidding any number in the support of b 3. Treat the first-round bonus won by bidder i as a sunk benefit, and note that since bidder 3 never bids eactly P not bidding is identical to bidding P Γs 3 for bidder i. By bidding Γ for s 3,, bidder i wins when s 3 < ; if bidder i has signal s i and believes s j, this gives epected payoff c + 3 + si + Γ c + 3 + si + c 3 + + s 3 3 s i s 3 3 s i 6 s 3 If s i, then this is the same for all, so bidder i is indifferent among all bids in the range of Γ and not bidding effectively bidding P, and is therefore best-responding by playing a mied strategy. As for bidder j, by those same calculations, if s i, bidding Γ would give an epected payoff of 3 s j 6 s 3 G, which is nonpositive for any and any s j, so j best-responds by not bidding. Finally, knowing the value of the security is c + 3 + s 3, bidder 3 solves the maimization problem ma b P,Γ {0} ln Gb c + 3 + s 3 b 0
Taking the derivative with respect to b gives gb Gb c + 3 + s 3 b c + 3 + Γ b b c + 3 + s 3 b This is positive when Γ b < s 3, or b < Γs 3 and negative when b > Γs 3 ; so bidder 3 s log-profit problem is strictly quasiconcave and solved at b Γs 3. For s 3 > s 3, c + 3 + s 3 Γs 3 c + 3 + s 3 c 3 + s 3 + s 3 s 3 3 s 3 s 3 s 3 > 0 so bidder 3 prefers bidding to not bidding; and for any s 3 s 3, c + 3 + s 3 P, so bidder 3 prefers not bidding to bidding. What we need from CG3 is for bidder j the one who is not the first-round winner to earn 0 payoffs which is true since he doesn t bid, and if P P, for player i to earn non-postiive e-ante payoffs. To see the latter, note that means s 3 c + 3 + s 3 P P c + 3 6ρ ; bidder i s second-round profit is therefore + 6ρ 3 s i 6 s 3 ρ and therefore even counting the first-round bonus, bidder i s profits are not positive. What if? For, the game is modified in the following way: First-round strategies are the same, but with P and P modified to be P c+ and P c + + ρ + ρ If bidder deviates in the first round but drops out at a price P P, leaving bidder as the first-round winner, CG is replaced by a drainage tract auction where s and s U0,. Bidder s strategy is to bid Γ s i c + + s if this is above P, and not bid otherwise. Bidder s strategy is to mi over not bidding and bidding in the range of Γ, such that the
distribution of his bids is if Γ G ep Γ dt if P, Γ c+ +Γ t t 0 if < P with the point mass at P corresponding to not bidding. If bidder deviates in the first round and wins when bidder drops out at price P, CG is replaced by a different drainage tract auction where s and s U0,. Bidder s strategy is to bid Γ 3 s c + + s + ρ s if s > ρ, and to not bid if s ρ ; bidder mies such that the distribution of his bids is if Γ 3 G 3 ep Γ dt if P, Γ c+ +Γ 3 3 t t 0 if < P If both bidders deviate in the first round, leading to either bidder winning at a price P < P or either bidder winning a price P > P, beliefs are s s ; if P < c +, both bidders bid c + in the second round, and if P c +, neither bidder bids in the second round. In the case where deviated but still dropped out at P P, losing the premium, bidder s strategy implies that a bid of Γ by bidder gives epected payoff c + s + Γ s so if s, bidder earns zero payoffs from any bid and is thus content to mi. Given bidder s mied strategy, bidder having already won the premium and believing that s, by bidding b, earns additional payoff G b c + + s b. Taking the log and then the first- order condition gives, which is positive when b < Γ s c+ +Γ b b c+ and negative +s b when b > Γ s, making Γ s a best-response when it is greater than P and making not bidding, equivalent to bidding P, a best-response when Γ s < P.
In the case where bidder won the first round at price P, the logic is similar, but now bidding Γ 3 earns bidder a second-round payoff of c + s + Γ 3 c + s + c + ρ s ρ so if s, any second-round bid gives the same payoff of ρ, eactly countering the win of the premium in the first round. Since bidder won the premium, not bidding is the same as bidding P, and gives the same payoff of ρ. Again given bidder s miing strategy, Γ 3 is a best-response for bidder. If bidder deviates, bidder epects to drop out of the first round at price P and earn positive payoff in the second round; by dropping out earlier, he would induce bidder to bid c + in the second round, so he would earn 0. So it is credible for bidder to stay in until P. If bidder deviates, then, he epects to either drop out before P, in which case he earns nothing; or to win the first round at P, in which case he wins the premium ρ but then earns payoff ρ in the second round. So deviating in the first round is not profitable for bidder. Second-round play on the equilibrium path is the same symmetric equilibrium as in the > cases. References. Engelbrecht-Wiggans, Richard, Paul Milgrom, and Robert Weber, Competitive Bidding and Proprietary Information, Journal of Mathematical Economics 983, 6-69.. Hendricks, Kenneth, Robert Porter, and Charles Wilson, Auctions for Oil and Gas Leases with an Informed Bidder and a Random Reservation Price, Econometrica 6 994, 45-444. 3. Milgrom, Paul, and Robert Weber, The Value of Information in a Sealed-Bid Auction, Journal of Mathematical Economics 0 98b, 05-4. 3