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On ring formation in auctions
Mathematical Social Sciences 32 (1996) 1-37
On ring formation in auctions W e r n e r G/ith a'*, Bezalel Peleg b "Institute for Economic Theory, Humboldt-University of Berlin, Spandauer Str. 1, D-10178 Berlin, Germany blnstitute of Mathematics, The Hebrew University, Givat Ram, 91904 Jerusalem, Israel Received March 1994; revised November 1995
Abstract It is assumed that first ring members bid in a preauction or knockout and that then only the ring's representative actively bids in the subsequent main auction. The rules must specify for every vector of bids who represents the cartel in the main auction, which transfers are paid to other ring members, who wins the main auction, and finally which price has to be paid. Most of the rules can be derived instead of being imposed by requiring envy-free net trades with respect to bids. Although they could be uniquely determined by an additional incentive constraint, we try to find out for all envy-free auction mechanisms whether they are profitable, i.e. every type of every cartel member gains from ring formation, and coalition proof. The case when the ring representative has to repeat his preauction bid in the main auction is analysed in more detail than the case when these two bids are independent. Keywords: Auctions; Cartel formation; Ring formation; Incomplete information; Envyfreeness
1. Introduction
Auctions and public tenders account for many economic trade activities of indivisible commodities. Whereas in an auction buyers compete for buying one or several units of a well-specified commodity, potential sellers compete for delivery in a public tender. Especially, in the public sector all major investments are organised in the form of public tenders and their rules are often legally specified (see, for instance, Gandenberger, 1961, who reports on rules of public tenders in * Corresponding author. 0165-4896/96/$15.00 t~) 1996 Elsevier Science B.C. All rights reserved PII S0165-4896(96)00808-6
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W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
the early sixteenth century). Since all theoretical results for auctions can be transformed in analogous statements for public tenders just by exchanging the two sides of the market, we can, without loss of generality, restrict our attention to auctions. For the sake of simplicity, we, furthermore, assume that there is one single object to be auctioned (see G/ith, 1986, for the obvious generalisation of auction rules to situations where more than one unit of the same product is offered for sale). A ring in an auction is a subset of the set of all bidders that agreed to submit only one essential bid, namely the bid of the designated winner. The importance of ring formation in auctions is obvious and partly well documented (see, for instance, Hendricks and "Porter, 1989, as well as the other studies of ring formation that will be discussed below). Of course, this bid might have to compete with the bids of non-ring members. It might, furthermore, have to exceed the reservation bid of the seller. Here it is assumed that the seller announces his reservation bid before the auction, which seems to be the usual procedure (Graham and Marshall, 1987, explore different rules). In the case of two bidders, the ring does not have to face bids of non-ring members. Actually, this is the main complication when going from the case of two bidders to 'ring games' where the ring has to compete with the highest bid of non-ring members whose value is determined by a given distribution. Our approach assumes that first, ring members run a preauction to determine the designated representative of the ring who either has to repeat his bid (as assumed in Sections 2-5) or can bid independently (Section 6) in the following main auction, whereas all other ring members must bid the seller's reservation value in the main auction. Except for the representative in Section 6, all bidders submit only one (essential) bid, namely ring members in the preauction and non-ring members in the main auction. The reason for first requiring the ring's representative to repeat his preauction bid in the main auction is that this imposes a natural constraint on the bidders' attempts to exploit the seller. Especially in cases where all bidders cooperate, independent bids of the ring's representative would result in complete exploitation of the seller and signal almost surely that bidders are cooperating. However, it is shown in Section 6 that the assumption that the preauction bid has to be repeated is no restriction of our principal approach. Actually, some of the results can be directly applied to the situation where the ring's representative can bid independently in the preauction and in the main auction. By imposing the axiom of envy-freeness with respect to bids, i.e. according to his bid no player prefers another's net trade to his own, we derive the following rules: the designated representative is the highest bidder in the preauction, and the auction winner is the highest bidder in the main auction. The price of the main auction has to be in the interval between the second highest and the highest bid in the main auction. The designated representative must equally distribute among all
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
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ring members an amount between what the second bidder in the preauction would have earned in the main auction if this is non-negative, and what he himself earns in the main auction. By imposing the assumption of independent weights, all mechanisms can be described as points (A, p) in the unit square where A defines the pricing rule in the main auction, similarly to Gfith and Van Damme (1986), whereas p defines the transfer payments of the designated winner to all ring members. It is shown that none of the possible A, p-mechanisms is incentive compatible and that only one satisfies underbidding proofness, i.e. every strategy that prescribes underbidding the own true value is dominated. For the case of independent and identically distributed true values and two bidders, who form a ring, we derive explicitly the unique symmetric equilibrium in monotonic and differentiable strategies, which is always coalition proof, i.e. no other ring could improve upon this result. The case of more than two bidders is analysed in the form of 'ring games'. In a ring game the winning bid in the main auction can be either the highest bid of all non-ring members or the highest bid in the preauction. We distinguish simple ring games where the highest bid of all non-ring members becomes common knowledge before the preauction. In complex ring games this value is not known when deciding in the preauction, and all ring members expect it to be chosen according to the same distribution. In other words: in simple ring games this distribution is the Dirac measure, whereas the distribution is more widely spread in complex ring games. The two rather special assumptions that there is only one ring and that the ring representative has to repeat his bid are finally avoided by introducing a communication structure, described by an undirected graph, and by allowing ring representatives to choose different bids in the preauction and in the main auction. The incentives that different A-pricing rules imply for ring formation have been analysed previously (see Fehl and Gfith, 1987; Graham and Marshall, 1987; as well as Graham et al., 1990; and Robinson, 1985). The analysis of collusive bidder behaviour in auctions by Graham and Marshall (1987) as well as by Graham et al. (1987) is more closely related to ours since these authors also derive the behaviour of all ring members instead of assuming some exogenously given agreement. These studies are, however, much more special since they focus on special pricing and transfer rules, i.e. on a specific (A, p)-constellation in the unit square. More specifically, Graham and Marshall (1987) design an ingenious, incentive-compatible mechanism according to which ring members in the preauction and bidders in the main auction can bid truthfully. But to avoid our non-existence theorem they have to pay a - in our view prohibitively high - price, namely the invention of an artificial institution, which they call a 'ring centre' and which is not modelled as a player. This serves only as a clearing institution that collects and distributes money within the ring. This ring centre makes lump-sum payments to ring members and collects payments from the ring's representative in
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the main auction if he wins. Although we acknowledge that the institution of a ring centre is an ingenious idea, we can hardly imagine how such an institution should be or can be implemented. From our analysis it also follows that the pricing rule, on which Graham and Marshall (1987) rely, is the most vulnerable one with respect to coalition proofness. It inspires attempts to offset the agreement by forming subrings. The problem, studied by Graham et al. (1987), is how a ring, which won the item in an auction, allocates it among its members via secondary auctions called 'knockouts'. Instead of allowing only one knockout they, furthermore, assume a nested subring structure, i.e. within the initial ring there is a smaller subring for which a smaller subsubring exists, etc. The idea of nested rings, which determine their representatives in larger rings by subsequent knockouts, is interesting, although the stylised Fact 7 of Graham et al. (1987) seems to be a very weak support. The authors suggest a rather detailed list of stylised facts (see also Graham and Marshall, 1987) without providing any evidence or references. McAfee and McMillan (1992) analyse ring formation in auctions from a welfare-economic point of view. Specifically, they first restrict themselves to incentive-compatible direct mechanisms, for which bidding truthfully is an equilibrium, by relying on the revelation principle. Within the class of such mechanisms they then determine those mechanisms for which the sum of a priori-expected profits by all ring members is maximal. Unlike in our study, coalition proofness is neglected by taking enforcement of the cartel agreement as granted, e.g. by an enforcing institution or by Folk Theorem-types of arguments (see McAfee and McMillan, 1992, p. 581). The authors distinguish weak and strong rings where transfer payments are (im)possible in (weak) strong rings. Most of their analysis assumes that all bidders are ring members. Incomplete tings are only discussed for the example when true values can assume only two levels. In our view, actually implementable mechanisms are not derived when using the revelation principle. Usually, the optimal mechanisms depend crucially on the given beliefs that in reality will change rather often and rapidly. Actually, the possibility that the mechanisms may depend on beliefs and the weaker notion of incentive compatibility (truthtelling is in equilibrium but is not necessarily the only undominated strategy) enable McAfee and McMillan (1992) to avoid our impossibility result. It is rather questionable whether a seller wants his sales procedure to depend on beliefs and whether bidders are willing to accept such dependencies. More importantly, the mechanism is assumed to depend on beliefs that are common knowledge, which raises the problem as to when and how this assumption can be satisfied. Thus, it seems justified to say that McAfee and McMillan investigate more the range of possible results than actually implementable mechanisms for ring formation in auctions. The remainder of this paper is organised as follows: in Section 2 we narrow down the set of mechanisms to the unit square (A, p) with 0 ~
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the possibilities for incentive-compatible mechanisms. Section 3 is devoted to the special case of n = 2 bidders for which we derive the general solution for all A, p-mechanisms and all information conditions satisfying the liD assumption of independently and identically distributed true values. Ring games are then the subject of Sections 4 and 5 before studying, in Section 6, the case where the ring's representative can bid independently in the preauction and the main auction. Some limitations of our study are finally discussed in our concluding Section 7.
2. An axiomatic approach We assume that a unique and indivisible commodity is to be sold by an auction with bidders i = 1 . . . . , n where n ~>2. To have something specific in mind, think of a unique piece of art or a building in a certain location. A ring is a non-empty subset C of N = { 1 , . . . , n}. Bidders in the complement (~ = {i ~ N: i,~C} of C will be called non-ring members or (ring) outsiders. We denote by c the number of bidders in C. Our analysis is based on the following basic assumption about the market decision process (Graham and Marshall, 1987, rely on a similar sequential model): first, all ring members i E C choose bids b,.(~>0). Only one member r E C repeats his bid and all other ring members bid the seller's reservation value in the following auction, when all outsiders j E t~ choose their bids bj(~>0). To have a convenient terminology, the first stage, when all ring members bid, is called a preauction, whereas we refer to the second stage as the main auction. Observe that every bidder k E N submits just one bid, b k. Thus, the decisions of both decision stages are summarised by the vector b = (b I . . . . , b,) of nonnegative bids. To determine the auction mechanism with ring C, the rules of the game must uniquely determine for all possible bid vectors b the following results: • the winner w(b) in the main auction; • the price, p(b), which the winner, w(b), must pay to the seller; • the winner, r(b), of the preauction, i.e. the designated representative of the ring C in the main auction; • the transfer payments t~tb)(b) of the ring's representative, r(b), to all his fellow ring members, i, with i ~ C and i ~ r(b). Thus, an auction mechanism with ring formation can be described by
/z = (N; C; w(.), p(-), r(.), (t~tb)(-) : i ~ C, i ~ r(b))), with N being the set of bidders, C the ring, and w(-), p(.), r(.), and t'(b)(-) for all i E C, i ~ r(b), being functions of the bid vector, b. In what follows we narrow down the class of possible mechanisms, /~, by imposing some intuitively convincing properties.
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Axiom 1. Envy-freeness with respect to bids. According to his bid, no bidder prefers another bidder's net trade to his own. Axiom 1 essentially rules out arbitrary generosity, i.e. better terms of trade for specific bidders. There is, however, a slight ambiguity when generalising the axiom as used in Giith (1986) to the situation at hand. Let us consider the case r(b) = w(b), i.e. the object is sold to the ring's representative. When deciding whether or not another ring member i ¢ r ( b ) prefers r(b)'s net trade, it is necessary to specify whether this includes also the price, which r(b) pays, or whether this decision has to be based on the price that i, as the ring's representative, would have to pay in case he wins the main auction. In the following, Axiom 1 will always be interpreted in the sense that ring members i ~ r(b) do not rely on the price r(b) pays, but on the price that they would have to pay as the ring's representative. Thus, the main implications of Axiom 1 are described by Theorem 1. For a mechanism Is, Axiom 1 implies the following requirements: (i) For all b, the winner, w(b), of the main auction is the highest bidder in the main auction. (ii) For all b, the sales price, p(b), must satisfy bw(b) ~ p(b) >I bz(b),
where bz(b) is the second highest bid in the main auction, i.e. the highest nonwinning bid. (iii) We denote by pi(b) for all i E C the sales price in the main auction if i was chosen as ring C's representative in the main auction: for all bid vectors, b, the ring's representative is the ring member r(b) ~ C whose surplus b,(b) --pr(b)(b) is maximal among all ring members. (iv) For all b the transfer payments t~(b)(b) are zero if r(b) does not win the main auction; otherwise, r(b) makes the same transfer payment f(b)(b) to all other ring members, i.e. r~(b)(b) = tr(b)(b) for all i ~ r(b), i ~ C, where f(b) satisfies
c
where bs(b) --pS(b)(b) is the second highest surplus in the preauction. Proof. Envy-freeness with respect to bids applied to the main auction implies
bw(b) -- p(b) >I 0 >I b i - p(b) for all bidders j with j ~ w(b) in the main auction. By adding p(b) we obtain:
bw(b) >~p(b) >I bj
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for all bidders j ~ w(b) in the main auction from which statements (i) and (ii) of Theorem 1 follow. Similarly, t~(b)(b)>I tT!b)(b)>! t~(b)(b) for all ring members i, i' with i, i ' ~ r(b) implies that all ring members must receive the same transfer payment from r(b). Applying Axiom 1 to the preauction yields, therefore, the condition
b,(b) -pr(b)(b) -- (C -- 1)t'(b)(b ) >i f(b)(b ) >! b i - p i ( b ) - (c - 1)/'(b)(b) or
b ,(b) -- P~(b)(b ) C
>>-tr(b)(b) >!
bi - pi(b ) C
'
for all i E C, i ~ r(b) .
The latter condition implies statement (iii) of Theorem 1 as well as statement (iv) for the case b,(b)~>p~(b)(b). If b~(b)>~pS(b)(b) does not hold, then the highest outside bid,
b c = max{b/: j E C } , satisfies bc>b~(b). For b¢>br(b) a positive transfer payment tr(b)(b) would obviously violate Axiom 1, since according to his bid, r(b) would prefer the net trade f(b)(b) to his own. For b,(b)>~ b c > bs(b) it follows from
br(b) -pr(b)(b) -- (C -- 1)f(b)(b) I> tr(b)(b) >! --(C -- 1)tr(b)(b) or
b,(b) -- p'(b)(b ) c
>I t'(b)(b) >i 0
that tr(b)(b) must be non-negative. Thus, we have proved statement (iv) for all possible cases bs(b) >I be, br(b) >~ b c > b~(b), and b c > br(b). Q.E.D. According to Theorem 1, the winner of the auction must be the highest bidder. This is the usual assumption in auction theory, which mostly is assumed without any justification. The minor ambiguity concerning the case of more than one highest bid in the main auction can be avoided by imposing anonymity. The winner of the preauction is the ring member whose bid would earn the highest surplus in the main auction. Another implication of Theorem 1 is that the whole transfer payment must be distributed equally among all ring members. The lower bound for this sum is what another ring representative could have won, i.e. bs(b) - pS(b)(b) if this is non-negative. The upper bound is the profit br(b) - p'(b)(b) of the ring's actual representative, r(b), in the main auction. It may cause confusion that it depends on the price pi(b) for all i E C who is chosen as the ring's representative, r(b). Obviously, this means that ring members must rely on an unambiguous prediction of the bids bj by all non-ring members. According to Axiom 1 alone, the prices pi(b) can be arbitrarily chosen in the
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range bw(b)>~pi(b)>I bzcb). So we cannot rule out by Axiom 1 alone that for two ring members i and ] we have pi(b)>pi(b), although b~ < bj. Such arbitrary dependencies will, however, be eliminated by our further requirements for mechanisms t~. Another possibility is, of course, to rely on the stricter generalisation of G/ith's (1986) axiom according to which envy-freeness within the ring is based purely on the price that the ring's representative, r(b), pays in the case when r(b) = w(b). Let us refer to this as Axiom 1'. Clearly, this will not affect statements (i) and (ii) of Theorem 1. Thus, the stronger implications of Axiom 1' can be described by substituting (iii) and (iv) in Theorem 1 by the following. Theorem 1'. For a mechanism tz, Axiom 1' also implies, in addition to (i) and (ii)
of Theorem 1, the following requirements: (iii') For all bid vectors, b, the ring's representative, r(b), is the highest bidder within C, i.e. b,tb) <<-b i for all i E C. (iv') For all b the transfer payments t~fb)(b) are zero for r(b) ~ w(b), otherwise r(b) pays the same amount trtb)(b) to all other ring members where t'~b)(b ) satisfies b'~b) - P(b ) >"t't°)(b ) >" max{o' bstb) c P ( b ) } Proof. When Axiom 1' is used instead of Axiom 1, the condition that r(b) does not prefer another ring member's net trade and vice versa becomes
brtb) - p ( b ) - (c - 1)tr(b)(b ) >- t'~b)(b ) >I b i - p ( b ) - (c - 1)t'(b)(b ) for all b with r(b)= w(b) and i E C. Clearly, this implies brtb)>~ b~ for all i ~ C and the proof of statement (iv') is similar to the proof of statement (iv) for Theorem 1. Q.E.D. According to (iii') the ring members obviously do not have to rely on expected and unobserved prices when deciding who should become the ring's representative and how much he must transfer to others. However, Axiom 1' assumes that every ring member i ~ w(b) bases his envy decision on the same price although he as a ring representative would have to pay a different price in the main auction. The latter reason is why in the following we rely on Axiom 1 instead of Axiom 1', and therefore on Theorem 1. In general, both the price, p(b), in the main auction as well as the same transfer payment, trtb)(b), for all ring members, can vary arbitrarily with b within the bounds given by part (ii), respectively (iv), of Theorem 1. We rule out such arbitrary dependencies of the sales price and the transfer payments by assuming that for all bid vectors b the sales price p(b) according to part (ii) of Theorem 1 is a constant convex combination of bw~b) and bz(b) and that, similarly, the transfer
W. G~th, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
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payment tr(b)(b) is a constant convex combination of br(0)--P'(b)(b) and max{0, bs(b) --PS(b)(b)}, i.e. of its bounds due to part (iv) of Theorem 1. Together with Theorem 1 this assumption allows us to express p(b) and t'(b)(b) by
p(b) = (1 - h)bw(b) + hbz(b),
with 0 ~
and
t,(b)(b) = 1 --Cp (b'(b) -- P'(b)(b)) + p max(O, b,(b) -- p~(b)(b)} with O~
0 the ring's representative, r(b), /s the highest bidder in the
preauction. Proof. Since br(b)--pr(b)(b)>~bs(b)--pS(b)(b) is equivalent to Abr(b)>~Abs(b), the claim follows for all A with 0 < )t ~< 1. For A = 0 we have b i - p i ( b ) = 0 for i E C and all bid vectors b, with i = w(b), and i as the ring's representative. Since this implies also that t'(b)(b)= 0, regardless of who is the ring's representative, Corollary 1 cannot be extended to the border case A = 0. Q.E.D. Up to now we have narrowed down the class of possible mechanisms p without exploring how the bidders will actually behave. To investigate bidding behaviour however, we have to specify how the bidders evaluate the object under consideration. Here we assume a private value auction, i.e. individual values Vk(>~O), which the various bidders k = 1 . . . . . n assign to owning the commodity under consideration. Furthermore, a bidder k may know his own true value v k but have only probabilistic beliefs concerning the true values ot of others. An axiom that would allow us to solve all auction games defined by Ix and the vector v = ( o t . . . . . on) of true values, regardless of the beliefs as to how other bidders evaluate the commodity, is as follows. Axiom 2. 'Incentive compatibility'. Bidding truthfully, i.e. bk = ok, is a dominant strategy, both in the main auction as well as in the preauction. Theorem 2. There exists no mechanism Ix that satisfies Axioms 1 and 2 when
weights are constant. Theorem 2 is proved in our working paper (G/ith and Peleg, 1995). The interested reader may obtain a copy from the authors. It is no surprise (see the more general result of Gibbard, 1973) that Axioms 1 and 2 are inconsistent. To avoid this inconsistency, Graham and Marshall (1987)
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have to introduce an artificial clearing institution called a ring centre. Since all payments within the ring C are transformed into payments to or from the ring centre, which is not considered as a player, our Impossibility Theorem 2 can be avoided. In our view, impossibility statements show that too much is required. We want to indicate that it is possible to axiomatically derive a unique mechanism p. = (A, p) ~ [0, 1] 2, if Axiom 2 in Theorem 2 is replaced by a weaker requirement. Axiom 3. 'Underbidding proofness'. For ring members the condition of incentive compatibility is replaced by requiring only one half of it, namely underbidding proofness, i.e. bidding bi < v~ is for all ring members, i E C, a dominated strategy. For ring outsiders the requirement of incentive compatibility is maintained. Theorem 3. The only mechanism Iz that satisfies A x i o m s 1 and 3 is defined by k=l=p. Proof. We have to show that A = p = 1 is the only mechanism satisfying Axiom 3. Clearly, for all ring members i ~ r(b) the transfer payment, trO)(b), can only be increased by overbidding vi and not by underbidding. Assume now that r(b) wins the main auction. Since his payoff is A V,(b) -- (1 -- a)b,(b) - Ab e - (c - 1)(1 - p) c (b'(b) -- b e ) - -
(C -- 1)p ~ max{O, b~(b) -- b e } ,
underbidding proofness requires that the coefficient -
1-A+c(C-
1)(1-p)
of b,(b) is non-negative, i.e. r(b) cannot increase his payoff by bidding a lower value. This, however, is equivalent to A[ 1 - c -cl ( l - P ) ]
~>1,
which can only be satisfied by A = 1 = p. Clearly, the mechanism /z defined by A = 1 = p is incentive compatible for ring outsiders and underbidding proof for ring members as required by Axiom 3. O.E.D. Although Theorem 3 suggests a unique mechanism/z, we do not confine our analysis to this result. Overbidding or underbidding the true value may compete with other properties. Especially, our initial assumption that ring C is formed can only be sustained if there exists no possibility to improve upon the result implied by C by forming a subring of C. In the next section we investigate first, for the
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special case of n = 2 bidders, whether or not underbidding proofness, i.e. A x i o m 3, c o m p e t e s with the requirement of coalition proofness, i.e. stability with respect to the formation of subrings. M o r e specifically, we investigate all mechanisms /x = (A, p) E [0, 1] 2 when for i = 1 . . . . . n the true value v~ of bidder i is i's private information. B e f o r e doing so let us consider the extreme situation when all true values vt . . . . , v, are c o m m o n knowledge. Without loss of generality we can assume that v~ is the highest true value in the ring C, and v z the second highest true value in C, whereas v c denotes the highest true value outside C. We neglect in the following the uninteresting cases v~ = v c and v~ = v z. The case of an overall ring C = N can be captured by v c = 0. If v 2 t> v¢, then an equilibrium implies b~ = b 2 ~> b i for all ] ) ~ C, where v t ~>b 1 = b 2 ~ > U
2 ,
since v~ - (1 - a)b~ - A v e c-1 - ~
~>
1-p C
c-1 c,
(1 - p)(b~ - (1 - )t)b, - A v e )
p ( b , - (1 - a ) b i - ) t o e )
(b, - (1 - h)b, - ) t o e ) + p- ( b i - (1 - h)b, - A v e ) C
is equivalent to v~ ~> b,.. In the case when v~ > v e > v 2, equilibrium bids b~ = b 2 < v e are impossible so that we have v l>~b l=b2>~max{v2,ve},
wheneverw(b)~C,
i.e. w h e n e v e r v e < v~. This shows that in the extreme situation where all true values ae c o m m o n knowledge, it cannot be decided by equilibrium analysis alone how profits are distributed within the ring C. W h e n e v e r v~ > max{v 2, v e } , m a n y distributions of profits can be supported by corresponding equilibria. Fortunately, we will not envisage such troublesome ambiguity when allowing for private information about true values in the sequel.
3. Bidding behaviour in the case of ring formation (n = 2) W h e n solving auction games explicitly, we rely on the I I D assumption of I n d e p e n d e n t and Identically Distributed true values. According to this assumption the true value v~ of bidder i = 1 . . . . . n is known only to bidder i, whereas all other bidders expect o,. to be determined by an independent chance move whose distribution F(-) is the same for bidders i = 1 . . . . . n. Owing to the a priori s y m m e t r y of all bidders, we can hope to find a symmetric solution, i.e. an
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
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equilibrium point prescribing the same bidding behaviour b~(v~)= t(vi) for all bidders i. The symmetric bidding strategy must specify a unique bid t(v~) by bidder i for all values vi, which his opponents j(¢i) consider as possible (see Harsanyi, 1967-68). If the seller has chosen a positive reservation value, then all bids b~ and true values vi are understood as the amounts by which they exceed this predetermined reservation value. Thus, we can assume b~ >10 and v~ >t0 for all bidders i = 1 . . . . , n. If for all bidders i = 1 . . . . , n the highest expected value v~ is finite, then, furthermore, we can renormalise the monetary unit in such a way that 0 ~
3.1. The special case of n = 2 Owing to n = 2, the only interesting ring is C = N = {1, 2}. If bidder i = 1, 2 with true value v~ expects the other bidder j(¢i) to use the equilibrium strategy b/(vj) = t(vj), his payoff expectation, Ev,(bi), depends on his own bid b i as follows: Ev~(bi) =
I [oi-(a-A)b~-P-~ t(o:)
(1-P)Abi]f(v/)do: 2
bi~t(O j )
(I-
t(vj) ] /(v j) dvj , J
bi
or
'(b,))
Evi(bi) =
t-1(bi)
pA
+-'~'-bi(1-
F(t-'(bi)))-~-
f 0
I
t-I(bi)
t(u/)f(u/)dv/
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
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From
d
db-'-~E~, (bi) = (vi - bi)f(t-l(bi))
dt-l(bi)
db-------~+ - ~ - F(t-t(bi))
(1-~)
=
O,
we obtain the inhomogeneous (ordinary) differential equation t'(vi) + h(vi)t(vi) =
g(v~) with -2f(v,)
h(v,) = ph - (2 - A)F(vi)
and
-2f(vi)o i g(vi) = ph - (2 - A)F(oi) "
Its homogeneous version P(vi) + h(vi)[(vi) = 0 has a left solution,
i,(v,) = e,(2~" ~-- a
"X -[2/(2-a)1
r(Vi) )
for all O<-vi
and a right solution,
i,(o,) = e,( F(v,) - 2
p _.AA)-{2/(2-A)1 '
f°r v~ < v ~ < 1'c" E R '
where v i is the critical v,. for which
pa 2 -
a -
F(v,)
holds. Owing to v* ~ [0, 1], this solution, which is not a solution to our problem, is neither monotonic nor differentiable. To determine a particular solution that solves our problem, we try to derive a solution of the form t(vi) = c(vi)/(v~), where the constant 6 of the homogeneous solution i(.) is substituted by a differentiable function c(.) of vv Since t ' ( u i ) - c ( v 3 t " (o,)+~'(vi)¢(v,)
and
r(v,) + h(v,)c(v,)i(v,) = g(v,) , we obtain:
c(vi)E'(v,) + c'(v,)E(v,) + h(vi)~(v,)i(o3 = g(v,) . Substitution of ]'(vi) = - h(vi)~(v~) therefore yields: c'(v,) - g(vi)
~(~,) •
Integrating c'(vi) for both ](vi)= il(vl) and [(v,)= :~r(Vl) yields a left solution, h(vi), and a right solution, t,(vi), including the coefficients ct and ~,, respectively. To make the solution t(vi) differentiable at v i we obviously must have tt(vT)= v i = t,(v i ), where we recall that
14
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
2f(v3(t(v,)
t'(v,)
-
v,)
= pit _ (2 - it)F(v,)
"
This boundary condition leads to
v~ + t(v,) =
(~ k2-/I
-
F(vi) ]
-
dx
'
for 0 ~ vi ~ v~
'
°' V i
v, -
:_--~-~/
\f(vi)
dr,
for v* <- v i <- 1.
v7
Owing to the definition of v*, the integrand on the right-hand side of the equation above is always non-negative and not greater than 1, so that t(vi) is well defined for all vi E [0, 1] and prescribes underbidding the true value in the range v~ > v* and overbidding in the range v,. < v*. Since f(.) is positive and continuous, it also follows that t'(v~) is positive for all v i ~_ [0, 1]. For the special case of a uniform distribution on [0, 1], the bidding function t(vi) is of the simple form pit 2 t(vi) = 4 - it + ~
vi"
The parameter p does not matter at all for the bidding behaviour in the case when it = 0, i.e. when the price p ( b ) in the main auction is solely determined by the highest bid b,tb). The intuition for this invariance result is, of course, that in the case when p ( b ) = b~(b) the designated ring's representative r(b), with b,(b) >I bs(b) , does not pay a lower price if all other ring members bid the seller's reservation value in the main auction. For p = 0, an increase of it from it = 0 to it = 1 implies a counter-clockwise rotation of t(vi) of the bidding curve. When it is positive, an increase of p causes a parallel upward shift of t(v~). The payoff expectation Eo, for bidder i's type v~, implied by the linear solution t(v~) for the special case of uniform beliefs, is it(pit + 1 - p)
Eo,=-~-+
2(4-it)
,
for a l l v ~ [ 0 , 1 ] ,
i=1,2.
Thus, in the case when it > 0 , even for vi = 0 the payoff expectation E~ is positive. Notice that the positive bid ti(0 ) for it > 0 implies the risk of losses. Since Eoi is positive for i t > 0 , this risk is, however, compensated by the gains of overbidding. For it = 0 we obtain Eo, = v ~ / 2 . Here, no bidder can ever lose money. If it = 0, then the payoff expectation Eo, does not depend on the transfer rule p. The other extreme pricing rule it = 1 implies 9
v7
1
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1 - 3 7
15
The formula of the equilibrium payoff expectation Eoi for the generol distribution F(vi) with carrier [0, 1] is derived in Appendix A. We do not try to discuss this much more complicated formula. Consider now the situation of a non-uniform distribution F(-) on [0, 1]. If v* = F - t ( p A / ( 2 - A ) ) E (0, 1), then the bidding function t(vi) assumes that A is positive. For A = 0 the bidding function t(vi) would always satisfy vi/> t(v~), as for the case of the linear distribution. For A > 0 one overbids in the range 0 ~
J
[ p - F(x) "~2 d x ~ . - r(oi) l
for 0 ~ Vi ~ V i '
{F(~)-p]2dx ~ F(vi) - P l '
for V *-
.
v, vi l= Vi
-
/-F-l(p)
By writing b~(tT,)=t(6"~) we can substitute 1/t t (vi), so that
t-l(bi) by tT~ and dt-t(bi)/dbi by
d X dbiEvi(bi) (vi-t(vi))~f(vi) + ~~--~- (1 - --~) F(vi). Inserting (t(
-
t'(~,) - (pA/2) - F(~.)(1 - ( I / 2 ) ) yields d
db i Evi(bi):
[-~-(I
A
-
-~) F(vi) ] •
vi-vi
•
This means that we rely on the bidding function b i ( v i ) = l(tTi), which assigns a bid t(tT~) to all values tT~E [0, 1]. This has to be distinguished from the necessary condition,
d 0 =~iiE°,(bi)lb,='(°, )~t'(vi)-'
(t(vi) vi)f(vi) - (pA/2) F(v,)(1 - (A/2)) ' -
for a local payoff maximum. We want to check whether local deviations from
t(v~) imply a payoff decrease.
w. Giith. B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
16
For t(Oi)> vi.a local deviation from t(v,) to t(ffi) implies also t(ffi)> ~7i, which is true only for ff~ < v *i. Hence (pAl2) - (1 - ( A / 2 ) ) F ( ~ ) tT~- t(ffi)
(,)
is negative. Thus, dEo,(bi)/dbi is positive for t7i < o i and negative for 17i>v~, which proves that the equilibrium bidding behaviour bi(v~)= t(v~) for bidders i = 1, 2 and all values v~ E [0, 1] with v~ < v,* is a local maximum. Obviously, a similar argument proves that b~(v~)= t(vi) describes also a local maximum in the range v~ > v *i • The same analysis applies also to v ~ - v~. Our results are summarised by
Theorem 4. There is a unique symmetric equilibrium in monotonic and differentiable bidding strategies, namely
v; for
O ~ v i ~ v
i ,
Vi
t(vi) =
Oi
\ F(v,) - 2--ea-z/
for v *~
It has to be expected that we can obtain the uniqueness result for the solution strategy t(.) by imposing much weaker conditions. For normal auctions, Plum (1992) has shown, for instance, that the requirements of monotonicity and differentiability by Gfith and Van D a m m e (1986) can be derived if we just require measurability of solution strategies.
3.2. Coalition proofness and profitability o f the ring formation for n = 2 We first define coalition-proof mechanisms in general before restricting them to the special case of n = 2. Let
G(S, . . . . .
U, . . . . .
On)
be an n-person normal form game. For every strategy vector g --- (st, • • •, s,) E S t × . . - x S, and coalition C C N = { 1 , . . . ,n} we define the normal form game G
=
by
U (s% = U,(sc,
(Uf)j
c)
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
17
for all s c= (sj)jec~ ×iEc Si, where gN\c= (Sk)k~N.k~C" We call GC(g) the Creduced game for .~. With the help of this notation we can define coalition proofness recursively: if n = 1, every equilibrium is coalition proof. For n > 1 we assume that coalition proofness is defined for all n ' < n : an equilibrium s * = (s~ . . . . , s : ) of the n-person game is coalition proof if (i) for all coalitions C C N, C ~ N, the strategy vector (s*) c is coalition proof in GC(s*), and (ii) there does not exist another equilibrium g = (gl . . . . . sn) satisfying (i) as well as Uj(g)> U~(s*) for all j E N. Thus, coalition proofness essentially relies on a consistent use of payoff dominance in the sense that all players under consideration are better off. For n = 2, coalition proofness implies that there exists no other equilibrium that payoff-dominates the equilibrium s* under consideration in the sense of the second requirement in (ii). Since for n = 2 every proper subcoalition C of N = {1, 2} contains only a single player, the very notion of a strategic equilibrium implies the coalition proofness of s c in GC(s*). Since we only accept symmetric equilibria in monotonic and differentiable bidding strategies, the uniqueness result of Theorem 4 implies that there can be no other such equilibria payoff-dominating the solution t(') described in Theorem 4. Thus, an immediate implication of Theorem 4 is as follows. Corollary 2. If we consider only symmetric equilibria in monotonic and differentiable strategies, then the solution described in Theorem 4 is coalition proof. If only two bidders are present, i.e. if n = 2, then an alternative to forming the ring C = N = {1, 2) is to bid competitively. We prove that ring formation is more profitable for every bidder i and true value vi than competitive bidding. Gfith and Van Damme (1986), who study competitive bidding by assuming the same solution requir.ements as we do, have derived the following bidding function: 0i
i(vi) = v i - J \F(vi)]
dx,
for all ViE[O , 1] and i = 1 , 2 .
0
Observe first of all that i(v,.) = t(v~) for all v i ~ [0, 1] in the case when A = 0 since this implies v* = 0. Notice, furthermore, that A = 0 implies that there will be no transfer payment from the ring's representative to the other ring member since abstention from actively bidding in the main auction has no effect on the price that r(b) must pay in the main auction. Thus, for A = 0 the preauction in the case of ring formation is strategically equivalent to the main auction in the case of competitive bidding. Thus, the problem of profitability arises only if the parameter h is positive.
W. Giith, B. Peleg / Mathematical Soc&l Sciences 32 (1996) 1-37
18
We assume now that A is positive. For the special case of the uniform density on [0, 1], the payoff expectation, 2
vi A(pA + 1 - p) Ev~ = -2- + 2(4 - A) of bidder i's type v i ~ [0, 1] in the case of ring formation depends crucially on the parameter h > 0 , whereas the corresponding payoff expectation, /~oi = v~/2, for the case of competitive bidding (see eq. (111.22) of Giith and Van Damme, 1986) does not depend on h due to the so-called equivalence properties of symmetric competitive auctions. Clearly, A > 0 implies Eo, > I~o, for all vi ~ [0, 1], which proves the profitability of all mechanisms/x = (h, p) E [0, 1] z for the special case of the uniform density. In Appendix B it is shown that this result can be generalised to general distributions F(.) on [0, 1] satisfying our assumptions: Theorem 5. All mechanisms p. = (h, p) ~ [0, 1] 2 are profitable, which for n = 2 simply means that no bidder type v i E [0, 1] can gain by inducing competitive bidding. For h > 0 , all bidder types v i ~ [0, 1] gain by ring formation; for A = O, ring formation has no effect at all, neither on bids nor on the payoff expectations of bidder types as well as of the seller.
4. A subclass of ring games (n > 2) If there are more than two bidders, i.e. for n > 2 , and if a ring C C N = ( 1 , . . . , n} with at least two members is formed, then refer to such an auction simply as a ring game. To simplify our analysis we rely on a given bidding behaviour of all non-ring members. Actually, it will be assumed that the highest bid, x = max{bj: j ~ N , j ~ C } , of all non-ring members is determined by a distribution G on [0, 1], which is common knowledge to all ring members and whose density is denoted by g. Thus, the only strategic agents of a ring game are the members of the ring itself. As for the special case of n = 2, it will be assumed that the true value v,~~ [0, 1] of each ring member i E C is private information and that all other ring members expect vi to be determined by a chance move according to a distribution F(') on [0, 1], which is the same for all bidders i E C and which has a continuous and positive density f(.) on [0, 1]. In other words: the beliefs within the ring C satisfy the IID assumption. One way to justify the given behaviour of ring outsiders is to assume that outsiders are unaware of the ring. To be more specific we assume, for instance, that the liD assumption holds with respect to all bidders and not only for ring members. Since ring outsiders are unaware of the ring, they therefore would bid
W. G~th, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
19
competitively, as analysed by Gfith and Van Damme (1986), i.e. the behaviour of all ring outsiders would follow the bidding strategy: oj f ( F ( r ) "~(n-l)'(1-~) t(vi) = v j - J \ F ( v i ) j dr, for all vj ~ [0, 1], j , ~ C . 0
Let y denote the maximal true value among ring outsiders, i.e. y = max{vj E [0, 11 :/" ~ N, ./,~' C}. Clearly, owing to the IID assumption for all bidders in N the distribution of y is G c ( Y ) = F ( y ) "-c '
where c is the number of bidders in ring C. This implies that the highest bid, Y
f ( F ( r ) '~ (,-t)/O-x) x = t(y) = y - j \ F---(-~,/ dr, 0
by ring outsiders is governed by the distribution Gc(x ) = G c ( t - l ( x ) ) ,
which shows that we can justify our assumptions for ring games in a consistent way. As before, we search for a symmetric and strictly monotonic equilibrium t: [0, 1]--->[0, 1] of the ring members, i.e. t(oi)=bi(oi) for all viE[0,1] and all ring members i E C, where t ( v i ) > t(6i), whenever vi > 6 i. If type v/ of bidder i E C bids b/and assumes that all other ring members j ~ C rely on t(.), then his payoff expectation is denoted by Eo,(b~). To define Eoi(b~) formally we introduce the following useful notation: y = max{oj : j ~ C and j ~ i}, z = max{vj
:
j E CX{i}, j ~ k for some k with v k = y}.
Clearly, for given true values the payoff of bidder i depends on the ordering of x, the highest bid by ring outsiders, as well as t(y) and t(z), the highest, respectively second highest, competing bid in the preauction. Let ~(y, z) denote the joint density of y and z. To calculate Eo,(b~) we distinguish the following nine cases: (1) b i >t t(y) >~t(z) >~x; (2) b i >>-t(y) >~x ~ t(z); (3) b i >>-x>! t(y) >t t(z); (4) t(y) >~b i >~t(z) ~ x ; (5) t(y) I> b i ~ x >~t(z); (6) t(y) >~x >~b i >I t(z); (7) t(y) >1t(z) >I b i ~ x ; (8) T ( y ) > ~ t ( z ) ~ x < - b i ;
W. Giith, B. Peleg I Mathematical Social Sciences 32 (1996) 1-37
20
(9) t(y) >I x >! t(z) >I b i. W h e n e v e r x is larger than b i and t(y), a ring outsider m a k e s the deal so that b i d d e r i's profit is zero. Relying on this case distinction, Eo,(bi) can be described as the sum: Eo,(bi) = I 1 + ' ' "
+ 19 ,
where y
t-l(bi)
f f [o,-(1-X)b,-Xx--a(c -
,l= f
1)
(pt(y) + (1 - p)b,)
z~O x=O
y~O
+
t(z)
a(c- I) c
t-1(b,)
]
x
g(x) dx~(y, z) dz d y ,
t(y)
a(c - 1) y=O
+
( p t ( y ) + (1 - p ) b , )
x=O
;,(c- a) ] c
x J ~(y, z) dzg(x) dx d y ,
b i t - l(x)
y
fir, - ( 1 x=O y=O
A ) b , - Ax
A(c c- 1 ) (1 - p)(b, - x ) ]
z=O
x ~p(y, z) dz dy dyg(x) d x , 1
t-l(bi)
t(z)
f f f
I, =
y ~ t - I ( b i ) z=O 1
15 ~---
A[obi+(1-p)t(y)-x]g(x)dx~o(y,z
) dz d y ,
x=O
bi t-t(x)
f f f
A
c[Pb,.+ ( 1 - p ) t ( y ) - x ] , ( y , z ) d z g ( x ) d x
dy,
y = t - l ( b i ) x=O z=O
I 16=
f
t(y) t-l(bi)
A
f
f
c (1-p)(t(y)-x)•(Y,z)dzg(x) dxdy ,
y ~ t - l ( b i ) x=b i z=O
I
17 =
f
Y
bi
f f*
y = t - l ( b i ) z m t - l ( b i ) x~O
C [pt(z) + (1 -- p)t(y) -- x]g(x) dxq~(y, z) dz dy ,
W. Giith, B. Peleg / MathematicalSocial Sciences32 (1996) 1-37 1
,.:
f
t(z)
+ (1 - p)t(y) -x]g(x) dx~o(y, z) dz dy,
f
y=t-l(bi) z = t - l ( b i ) xffib i
1
/9:
y
21
f
t(y)
f
/-I(x)
f
A__C(1--p)(t(y)--X)~O(y, Z)dzg(x)dx dy.
y=t-I(bi) x=b i z=t-I(bi)
Unlike in the special case n = 2 where we allowed for general distributions F(.) on [0, 1] with positive and continuous densities f(.) on [0, 1], we mainly rely on the linear distribution F(vi) = v i E [0, 1] when studying ring games. As shown for n = 2, the results for the uniform density can be generalised in a straightforward way, but at the cost of highly complex formulae that do not lend themselves to easy interpretations and illustrations. It will become clear in what follows that the special case of the uniform density already requires quite complicated considerations. For the uniform distribution, cartel outsiders j , ~ C bid according to n-1 t(Vs) = ~
vs,
for all vs E [0, 1] and j , ~ C .
Owing to n-A t-i(X) =-~-T_1 x , we therefore have
Go(x) = L-F:--f_1x ) for all x E [0, ( n - 1 ) / ( n - A)] and all rings C CN. In what follows we mainly investigate ring games where all ring members i ~ C expect another ring member j's true value oj to be chosen according to F(oj)=vj ~ [0, 1] and where for all rings C C N all ring members expect the highest bid x by a ring outsider to be chosen according to Gc(x ) as defined above.
5. The solution of ring games
In addition to monotonicity, which has already been imposed to guarantee that the symmetric equilibrium bidding function t(.) is invertible, we require, as always, that t(-) is differentiable. In Subsection 5.1 we first derive the differential equation for t(.). Subsection 5.2 concentrates on the uniform distribution and
C=N.
w. Giith, B. Peleg / MathematicalSocial Sciences 32 (1996) 1-37
22
5.1. The necessary condition for the best response (n ~>3 and 2 <~c <~n) Let N = { 1 , . . . , n } with n~>4 be the set of bidders and, without loss of generality, let C = {1 . . . . . c} with 3 <~c
y
t(y)
A(c - 1)
f ff
,,=
y=0
--U-(or(y)
z=O x=O
A ( c - 1) ] + (1 - p)b,) + - - - - b - ' ~ x j g(x) dx~p(y, z) dz d y , we have dJl/db i = Jll + J12, where y ~(y)
t-l(bi)
A(c - 1)(1 - p).]
,,,=- [1-,÷
ff
f y~O
z=O
g(x) dxtp(y, z) dz dy
x=O
and
dt-~(b~) Jl2-
db i
t-t(bi)
f z=O
bi
_h(c_ -
f [v,-(1-X)b,-Xx
C
1) bi +
A(c- 1) x ] C
x=0
x g(x) dx~p(t-l(bi), z) dz. Furthermore, dl3/db i = J3t + J32, where bi t - 1(£)
J3x=-[
1-A+A(c-1)(1-p)]c
f
f
x=O y=O
and t-l(bi) I"
J
]32
y=O
Notice that
y I"
J (o i - b,)g(bi)~p(y, z)dz dy. z=O
Y
f q~(y, z) dz dyg(x) dx g=0
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37 bi
t-l(bl)
y
Jll +J31 = - [ l - A + x =0
y=0
z =0
t-t(bD
A(c-1)(1-O)
= - [l-A+
f
]
(c-1)(F(Y)f-2f(y)dy
y-O
A(c - 1)(1 - p) ] G ( b i ) ( F ( t _ , ( b , ) l f _ ,
= - [l-A+ J12
C
dt-l(bi) db i f ( t - l ( b i ) ) ( c - 1)(F(t-'(b,))) c-2
Vi --
>(
1 -- C
b,
G(b,) - c
xg(x) dx x=O
, _1
and
J 3 ~2 (ui
- 1(b,))) c - 1
- b,)g(b,)(F(t
.
The sum I 4 + . . . + 19 can be expressed as J4 + -/5 + J6, with 1
y
t(y)
f f
J4 =
f
(l-p)A(t(y)-x)g(x)dxtp(y,z)dz
dy,
y = t - l ( b i ) z=O x=O 1
t-I(bi)
bi
f f f- -(bi-x)g(x)dx (y,z)dzdy, y=t-l(bi)
z=0
x=0
and
f
=
1
y
t(z)
f f (t(z,-x)g
yffit-l(bi) z ~ t - l ( b i )
xffiO
Differentiating these expressions yields A(c - 1)(1 - p ) c
d J4
dbi ×
b,C(b,)
dt-l(bi) db i f ( t - t ( b i ) ) ( F ( t - l ( b ' ) ) ) c - 2
xg(x) ax
-
,
x=O
d~ db i
Ap(c - 1) (1 - F(t-l(bi)))(F(t-l(bi)))C-2G(b,) c
23
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
24
+
Ap(c - 1)(c - 2) dt-l(b,) c db~ (F(t-t(bi)))c-3f(t-'(bi)) biG(b,) -
x (1 -
x f ( t - ' ( b , ) ) ( F ( t - ~ ( b , ) ) f -2
xg(x) dx
biG(bi) -
- Ap(c - 1) dt-~(b,) c db i
xg(x) (ix x=O
and
d J6 db,-
Ap(c - 1)(c - 2) dt-~(bi) c db i (F(t-~(bi)))c-3f(t-'(b~)) × ( 1 - F(t-'(b,)))
b,G,(b,)- .
xg(x) dx
x=0
We rewrite dEo,(b~)/db ~ in the form d d -1 db, E°, (b') = A(bi, v , ) - ~ i t (b,) + B(b,, v,), where the functions A(b~, vi) and B(bi, vi) are defined as follows:
A(b,, vi) = (c - 1)(F(t-l(bi))f-2f(t-l(bi))G(b~)(vi
- b~) ,
S ( b i, vi) = - (1 ---Ac) V(bi)(F(t-l(bi))) c-' + (o i - bi)g(b,)(F(t-l(bi))) ¢-1 +
p(c -
I)
c
(F(t-l(bi)))C-2G(b,).
For a local maximum of Ev~(bi) at the equilibrium bid bi=t(v~) dEo,(t(vi))/db i = O. Hence,
dt(vi) dvi
we need
A(t(vi), v,) B(t(vi), vi) '
or
dt(vi) dv i - Ap(c1-) c
(c - 1)f(vi)G(t(oi))(t(vi) - v,) G ( t ( v i ) ) - F(vi)[(t(vi)-v~)g(t(v~))+ (1---Ac) G(t(v~))]"
For the case c = 2, which we have excluded so far, the necessary condition for a locally best response is obtained by substituting c by 2 in the formula above. This result for the special case c = 2 can be derived in a similar way, but more simply. For C = N we would have, of course, G(x) = 1 for all x with 0 ~
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
25
d t ( v i ) = n ( n - 1 ) f ( v i ) ( t ( v i ) - vi) dv i A p ( n - 1) - F ( v i ) ( n - A) '
which, in the case of n = 2, coincides with the differential equation derived in Subsection 3.1). In the case of uniform distributions, we have F ( v i ) = vi for all vi with 0 ~
n-l~
'
f°rO<~t<~(n-1)/(n-A)
for t/> ( n - 1 ) / ( n - A). Thus, the differential equation has the special form: dt(vi) dP i
c(c - 1 ) t ( v i ) ( t ( v i ) - vi) = X p ( C --
1)t(Vi) -- Vi[C(t(Vi) -- v i ) ( n -- c) -k (c -- A)t(Vi) ] '
for 0 ~< t ~< (n -- 1)/(n -- A) and dt(vi) do i
c(c - 1)(t(vi) - vi) - Xp(c-1)-vi(c-
for l > - t > - - ( n -
A) '
1)/(n " A ) .
5.2. T h e s o l u t i o n f o r the s p e c i a l cases
In what follows we derive and discuss the solution of special cases like special rings C, or special distributions F(-) and G(.). We will often rely on techniques that have already been used when analysing the special case n = 2. In such cases we sometimes mention only the main idea of the proof instead of proving everything in full detail. 5.2.1.
T h e g r a n d ring, C = N
Proceeding as for n = 2, the case C = N for n > 2 can be solved by first deriving the solution of the homogeneous differential equation: di(vi) dv i
n(n - 1)f(vi)i(vi) - Ap(n - 1) - (n - A ) F ( v i ) "
and then applying the variation of the constant technique. This yields: Theorem 6. I f F ( v i ) = v i f o r all 0 <~ v i <~ 1, i.e. in case o f the u n i f o r m d i s t r i b u t i o n , then the s o l u t i o n is o f the s i m p l e f o r m : n2-n n-1 t(vi ) = n 2---1"-~ vi + ~
pA ,
f o r all 0 <~ vi <~ 1 .
26
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
Proof. As in the proof of Theorem 4, we can show that the homogeneous differential equation,
dE(o,) dv i
n(n - 1)i(vi) )t)v i '
- (n-1)Ap-(n-
has no monotonic solution. Thus, the solution, described by Theorem 6, is the only one that solves dt(v,) dv i
n(n - 1)(t(vi) - vi) - (n - 1)Ap - (n
-
t~)v i '
and satisfies the boundary condition t ( v * ) = v *i, where v * = Ap(n - 1)/(n - A). To prove that the differential equation above describes a payoff maximum, we can show that for every v~ E [0, 1] there exists ~ > 0 such that d I>0, dbiE°, ( t ( ~ ' ) ) = . 0,
forv~-~
For the uniform distribution, where n
6]
db, E°, ( t ( 4 ) ) -
-
x
n
-
-
6 i - t(ff/)
o,)
'
it follows from t7i < v i < v* that d E ~ , ( t ( S i ) ) / d b e is positive owing to t(St) > t7i in the range tT~< v,*. Other cases can be checked similarly. Q.E.D. To check the profitability of the ring C = N for the uniform distribution, we have to compute the profit expectation Eo~(A, p) implied by the solution t(v~) of the ring game and to compare it with the payoff expectation I~.o,()0 in the case of competitive bidding. As shown in detail in Appendix C, we obtain the following result for C = N and the uniform distribution: n (n - 1)A2p A(n - 1) , Eo,(A, P) = v__2t+ + n n(n 2 - A) n-~-2 - ~) (n - 1 - p ) . As shown by Giith and Van Damme (1986) the corresponding result for competitive bidding is l~o,(A) = v ~ / n . Theorem 7. F o r C = N a n d the u n i f o r m distribution F(v~) = v~ f o r all v i ~ [0, 1], the ring f o r m a t i o n
is p r o f i t a b l e in the s e n s e o f
Eo,(a, p) > t~o,(A) ,
for A > O,
Eo,(A, p) = t~o,(A) ,
f o r A = O.
The coalition proofness of the overall ring C = N and the uniform distribution
W. Giith, B. Peleg I Mathematical Social Sciences 32 (1996) 1-37
27
F(vi) = vi for all 0 ~
6. Ring formation with independent bids Although our results are not complete by far, let us briefly investigate a modified version of our basic model, which allows that the ring's representative can choose his bid in the main auction independently of his bid in the preauction: clearly, this will lead to an even more complete exploitation of the seller (in case of an auction). In the case of a grand ring containing all bidders, the ring's representative, being the only essential bidder in the main auction, would, for instance, simply bid the seller's reservation price. We, of course, wonder how such a grand ring could remain undetected. Nevertheless, we think it worthwhile to explore the model where the bid in the preauction does not restrict the bid in the main auction. Here we do not restrict ourselves to situations where just one ring can form. By an (undirected) graph we impose a communication system among bidders. The components of the graph partition the set of bidders in such a way that all members of the same component can communicate with each other, whereas this is impossible for two bidders in different components. Thus, it is natural to assume that only components can form as rings, which, of course, allows that several rings can form. Our analysis is based on the following basic sequential decision process for a given constellation of rings C ~. . . . . C m with C* tq C j = 0 for k, j = 1 . . . . , m and k #/'. First, in the preauction, all ring members i E U "]=1 C j choose a bid bf(~
28
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
different bids b p, and br for a ring's representative r and by allowing for more than just one ring. An auction mechanism must specify for each vector b= (bi)izu?,l cJ. o~i=,J fo~,om, ~-~ . . . . . . . who is the winner, w(b), in the main auction, i.e. who buys the object under consideration, and which price, p(b), bidder w(b) has to pay. For the preauction, the auction mechanism must determine for j = 1 , . . •, m, and for all possible bid vectors b ~ = (bi)i~cJ P of ring C ~, the ring's representative, r~(bi), whereas the transfer payments t~ of r j to his ring partners (for not actively bidding in the main auction) i ~ C j with i # r j can depend both on b i as well as on b, i.e. t~ = t~(b/, b). Thus, an auction mechanism p. can be described by = [N; c ' . . . . .
cm; w(.), p(•);
...... ] ,
where N is the set of bidders, C ~. . . . . C " are the rings that have been formed, and w(.) and p(.) are the rules of the main auction. Whereas the ring's representatives r j are only determined by the respective bid vector b j, their transfer payments are allowed to depend on b i and b, since what happens in the main auction can influence the transfer payments, but obviously not who will bid in the main auction. The obvious implications of Axiom 1 are described by what follows• Theorem 9. For a mechanism ~, Axiom 1 implies: (i) For all bid vectors b in the main auction, the winner w is the highest bidder and the price must satisfy b w ~ p >~bz, where
b ~ = m a x ( b'' ffi~ Cj Ck Or j = r ' fOrsOme j =
.....
m, j # w }
,
i.e. the price has to be in the interval between the highest and second highest bid in the main auction. (ii) For all rings C j, with j = 1 , . . . , m, the ring's representative, r j, is the highest bidder of C j in the preauction; his transfer payment is equal for all i E C ~, with i ~ r j, i.e. t~ = t j for all i ~ C j with i ~ rJ; this transfer payment, t i, is zero in the case o f r i ~ w , whereas otherwise it has to satisfy (bP,j-p)/cJ>~tJ>~ max{0, (b~j - p ) / c ~ } , .where c ~ is the number of members of ring C i and b~j = max{b/p : i E C I, i ~ r t} denotes the second highest bid in ring C~'s preauction. Again, it is assumed that for all bid vectors b in the main auction the selling price p is a constant convex combination of bw and b z, the highest and second highest bid in the main auction, and that for every ring C j with j = 1 . . . . . m the transfer payment in the case of r j = w is the same constant convex combination of
IV. Gath, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
29
b re, - p and max{0, b~j - p }. Thus, the class of mechanisms/x is isomorphic to the unit square: {(h, p): 0 ~< A, p ~< 1} = [0, 1]2 ,
where A denotes the pricing rule p ( b ) = (1 - A)b w + Ab~ for the main auction, and p, the transfer rule: ti(b) =
[bP,i - p ]
+
max{O, b ~ j - p } ,
if r j = w .
If rings have formed, then the game is as described, where we allow for all mechanisms/z E [0, 1] 2. Whether or not a certain ring C i forms will be decided at a previous stage where all members i E C j have to accept the ring C j and where only those rings C j can be formed whose members are able to communicate with each other. To formalise the initial stage of strategic ring formation, we first introduce the notion of (undirected) graphs by which it will be specified with which bidders in N a given bidder can communicate. A graph G is defined by a set of vertices which will be the set N = {1 . . . . . n} of bidders for the case at hand, and a set E of edges. Formally, the set E of edges of the graph G = (N, E) is a relation on N, i.e. E C N x N: (i, j) E E then means that the graph contains an edge connecting i and j. Since we restrict ourselves to undirected graphs, the relation E on N is symmetric, i.e. (i, j) E E implies also (j,i)EE. We consider now two vertices i, j ~ N. We say that i and j can communicate with each other if there exists a sequence {k ° = i, k I . . . . . k "-1, k ~' = j } C N of vertices k ~, which starts with k ° = i and ends with k ° = j such that (k t, k t ÷~) ~ E for all 1 = 0, 1 , . . . , v - 1. If i and j can communicate, we write i - j . Clearly, defines a relation on N, i.e. ~ C N × N, which is reflexive ( i ~ i for all i E N), since we assume that all loops (i.e. edges connecting a vertex to itself) exist, symmetric (i ~ j implies j ~ i), since we can reverse the sequence, and transitive (i ~ j and j - k implies i - k), since we can link the two sequences connecting i and j and j and k, respectively. For any i E N we define by E(i) = { j E N : j ~ i } the subset of vertices in N with which i can communicate. We call E ( i ) the component of vertex i ~ N . Clearly, the set N is partitioned into components where vertices in the same component can communicate, whereas vertices in different components cannot. For our analysis of ring formation it will be assumed that there exists a graph G = (N, E) whose components are the possible rings C ~, . . . , C " . Whether or not a component C j of G forms and organises a preauction is the result of what follows. Initial stage. 'Strategic ring formation'. Let G = (N, E) be a graph with com-
W. Giith. B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
30
ponents C ~. . . . . C ' . For j = 1 . . . . . m the component C j becomes a ring if all members i E C j unanimously (and simultaneously) vote for this result, otherwise the component C i does not form a ring, i.e. all members i ~ C j abstain from ring formation. Throughout our analysis it will be assumed that although all members of all non-trivial components of graph G -- (N, E) consider whether or not to form a ring, they are completely unaware of other attempts to form a ring. In other words, every representative of a ring will think that he is the only one representing a ring (of more than one bidder). As a consequence, every representative expects the other bidders of the main auction to behave competitively. There is a need to justify such an assumption, especially to specify information conditions supporting it. One way is to assume that every bidder i ~ N knows only his own component and the set N. For bidders j E N with j ~ E ( i ) he believes that their component E ( j ) is given by E ( j ) = {j}, i.e. E ( i ) is, in bidder i's view, the only possible ring. Another consistent way to justify such an assumption is that every bidder thinks that only his component is capable of (illegal) ring formation. In the main auction, a bidder does not learn that fewer bidders are active, since bids are made privately (ring members abstaining from bidding in the main auction always bid the seller's reservation value). Thus, there can be no signalling of the rings. This completes the description of our model of ring formation based on a communication structure G = (N, E). In what follows we investigate only the possible mechanisms /z E [0, 1] 2 for private value auctions satisfying the IID assumption for the special case C ~ = N . For the special case of C ~ = N , our previous results already determine the solution. Let us assume that n I> 3 and F ( v ) for v E [0, 1]. Because of C = N we have p(-) - 0 and tJ(bp) = 1 - p n
p
b'+n
pb p
s,
where b p = (b p . . . . , b p) is the bid vector in the preauction whose highest and second highest component is b p, respectively b p. Let t(.): [0, 1]---~ [0, 1] denote the symmetric and monotonous bid function for the preauction. Relying on y = max{min(v k, or}: k, 1 = 1 . . . . . n; k ~ i ~ l ~ k } bidder vi's payoff expectation Eo,(b~) when bidding b,. must be defined for the cases that follow. For b i >>-t(y) we have t-l(bi)
E°, (bi) =
f
y=0
for t ( y ) >i b i >~ t(z):
I vi - ~n -n- 1 ( p t ( y ) + (1 - P)bi) ] (n - 1)y "-2 dy ;
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37 1
fI
Eo,(bi)=
31
t-l(bi)
(n - 1)(n - 2) n
[pbi + (1 - p ) t ( y ) ] z "-3 d z dy ;
yffit-t(bi) z=O
a n d for t(z) >- bi: 1
f
E~' (bi) =
y
f,.a, n2, n
[pt(z) + (1 - p ) t ( y ) ] z "-3 d z d y .
y=t-l(bi) z=t-I(bi)
These three integrals are identical to those that determine the solution in the case of C = N and A = 1 for the previous model. Thus, the solution for C = N, n ~ 3 and F ( v ) = v E [0, 1] for the new model is n 2 -- n n - 1 no~ + p t(vl) - n 2 - 1 vi + n 2 -"----~p - n +'-------~"
By Theorem 7 this equilibrium strategy is, furthermore, profitable, i.e. for all values o~ E [0, 1] of all bidders i = 1 . . . . . n it is true that Eo,(b~ = t(o~)) is greater than the payoff expectation ( v i ) " / n resulting from competitive bidding (Giith and Van Damme, 1986).
7. Conclusions
Although our results for general ring games are still limited, some important policy conclusions are already possible: whereas the only mechanism/~ satisfying Axioms 1 and 3 is defined by A = 1 = p, the opposite pricing rule A = 0 seems to provide the worst prospects for ring members. Since A = 0 implies b i = p i ( b ) for every cartel member, this pricing rule does not allow for positive compensation payments t~b)(b). Furthermore, in the case of A = 0, ring formation has no effect whatsoever on profit expectations in the case of overall rings C = N, whereas A > 0 implies positive payoff incentives to form an overall ring. Thus, the long historical tradition of the A = 0 pricing rule, which in public tenders organised by German public authorities can be traced back to the sixteenth century (see Gandenberger, 1961), seems to be well founded on experience, i.e. that this pricing rule is more immune against ring formation. Our results are, however, still incomplete, since we have not yet fully explored general ring games with C ~ N and n >~3. Especially, we have not been able to solve generally the differential equation derived in Subsection 5.1). Furthermore, only the special case When the ring representative can choose different bids in the preauction and in the main auction, has been solved. In future research we will try to overcome these limitations.
W. Giith, B. Peleg I MathematicalSocialSciences32 (1996)1-37
32
Acknowledgements
We gratefully acknowledge the comments of the referees as well as the advice by Herve Moulin. This paper was written when both authors were enjoying the inspiring atmosphere and hospitality of the CentER for Economic Research at Tilburg University, The Netherlands. Werner Giith's stay at CentER has been made possible by a generous Alexander von Humboldt Award via NWO (the Dutch Science Foundation).
Appendix A: The equilibrium payoff expectation for the general distribution
F(vi) Substituting
t(vi)
for v; and v~ = t-l(bi) in the payoff function Eo,(b;) yields:
]
1 - p A)t(vi) Eo, = [v, - (1 - ;t +---~
F(vi)+'-~'(1-F(vi))t(v;)
vi
_ ~
pA2 f
1
(1
t(y)dF(y)+~
f t(y)dF(y)
0
vi
=vf(vi)+ [P~-~-(1-2) F(°i)] t(vi)+ 1
2
vi
vi
0
This formula, together with u~ f l- p a l ( 2 - A)- F(x) ]zz(z-,~) t(v,) = v, + L.P~-7~- ~ -- F~D. J ~, vi 1
1
v*i
V*i
L p,a/(2 - a ) - F(y) J Vi
Vi
0i 1
Oi
f t(y)f(y) 0
Y
f jr LF(y) - [ p M ( 2 - A)]
_
Oi
dy=
dx/(y) dy
Y
Ui
dx/(y) dy,
01
f yf(y)dy+
f f L~---A)rPA/(2-A,-F(X)F(y)j]z't2-x)dxf(y)dy,
0
0
Y
for o; <~v* and the corresponding formula for vi ~>v,*., determines E~, for the symmetric equilibrium in which the two bidders, j = 1, 2, rely on bj(oj) = t(ej).
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
33
Appendix B: Proof of Theorem 5 We rely on the notation and results of Section 3, including Subsection 3.2. The monotonic bidding function, derived in Giith and Van Damme (1986), is oi
f (F(x) ~ 1/O-a)
t(vi)
"~- V i
dx,
-- J \ F - ~ i ) ]
v i E [0, 1],
for all
A E [0, 1),
i = 1, 2 ,
0
which is the solution of the differential equation f ( v i ) ( V i -- i(Vi) )
i'(vi) -
(1 -
A)F(vi)
derived from the necessary condition for a local maximum of i-I(bi)
g~,(b,) = f
[v, - (1 - h)b, -
M(x)]f(x) dx.
0
Owing to oi
f
Oi
i(x)f(x) dx = i(v,)F(vi) - f F(x)f(x)(x - t(x))
0
dx
0 0i
= i(v,)F(v,)
1 -1 A f f ( x ) . x d x
+~
if
Oi
i(x)f(x) dx,
0
we obtain: oi
.f
oi
If
i(v,)F(v,)+-i-~
i(x)f(x) dx = " ~ X
0
xI(x) dx ,
0
and therefore Vi
Vi
A f i(x)f(x)dx +(1-A)i(v,)F(v,)= f xf(x)dx. 0
0
Hence, Oi
Eo,(A ) =
viF(v,) - f xf(x) dx, 0
since in equilibrium we have i - l ( b ~ ) = vi and
bi =
i(vi) and therefore
W. Giith, B. Peleg / Mathematical
34
Social Sciences 32 (1996) 1-37
oi
~.o,(x) = f Iv; - (1 - x)~(v,) - Ai(x)]f(x) dx 0 vi
=
x f ;(xly(xl x.
(1-
0
The fact that t~o,(I ) does not depend on ,~ implies the well-known equivalence of pricing rules (see Gfith and Van Damme, 1986, for a discussion). We want to compare l~o,(;t) with the equilibrium payoff expectation: Of
Eo, (A, p) = viF(oi) - (1 - A)t(vi)F(vi) - ~
f t(x)f(x) dx 0
(1 -p)a 2 t(vi)F(vi) 1
+ P - ~ t ( v i ) ( 1 - F ( v i ) ) + -(1- 2p)A f t(x)f(x)dx vi
0 1
+
t(vi) +
2
t(x)f(x) dx, 0i
for ring formation. We observe that vi
2 - A (t(vi) - t(0)) -
t(x)f(x) dx 0
oi
oi
0
0 vi
=-t(oi)F(oi)-
f (~
~¢x,)t'(x)dx
0
= - t(vi)F(°i)
2--~t
I(x)(t(x) - x) dx 0 0i
=
-
t(v,)F(v,)
2 -
Ui
t(x)I(x) dx +-ff-2--2 xI(x) dx,
x 0
0
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
35
owing to t'(x) = - h(x)t(x) + g(x), where the functions h(.) and g(-) are defined in Subsection 3.1. Hence, Pi
2
Oi
xf(x) dx = - t(vi)F(vi) - 2 - A.
2-A 0
t(x)f(x) dx 0
+ 2 ~ A (t(vi) - t(0)), or equivalently, Vi
Vi
A ( 1 - 2 ) t ( ° i ) F ( v i ) - 2 f t(x)f(x)dx+ - ~ (t(v,)-t(O)).
- f xf(x)dx=0
0
Thus, the inequality l~v,(A) ~ Eo,(A, P) is equivalent to vi
A
viF(v,)- (1-2) t(vi)F(vi)--~ f t(x)f(x)dx + ~ _
(t(vi)
_
t(O))
0 vi
<~v,F(vi)- ( 1 - 2 )
t(vi)F(vi)-P-~
pA f t(x)f(x) dx +--~-t(vl) 0
1
+ ~(1 - p)A f t(x)f(x) dx, vi
or to the inequality 1
(a-p)X f 2
vi
x
t(x)f(x)dx+-~(1-p)
vi
;
t(x)f(x)dx+
t(0)>~0,
(*)
0
which is always satisfied due to 0~A, P, vi ~< 1 for i = 1, 2. Clearly, t~,(A)= Evi(A, P) for A = 0, whereas by (*) we have Eo,(A,p)>I~,,(A),
for all0<-p,v~<-l,
when A is positive. This proves Theorem 6 also for the general distribution f(-) that satisfies our regularity conditions. Q.E.D.
Appendix C: The equilibrium payoff expectation for C = N and the uniform distribution
To prove Theorem 7, we show that Eo,(h,p)
v~+ (n-1)A2p = n
A(n-1) (n-l-p) n(n ~ - ~) + n(n ~ - X)
:
IV. Gfith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
36
for C = N and the uniform distribution F(v~) = vi for all vi E [0, 1]. We denote by a the coefficient of v~ and by /3 the constant of the solution function t(v~), described in T h e o r e m 8, i.e.
n(n or-
1) n2_A
n - 1
and 1 3 = ~ A p .
For this special case, Eo,(A, p) can bc described as Eo,(A, P) = K I + K 2 + K 3 , where Oi
Kt= ~( [vi-(1-A)(otv,+/3 )
a(n-1)n
(p(oty+/3)
y=O
+ (1 - p)(av, +/3))] (n - 1)y "-2 d y , 1
/ .~
K~ = (n - 1)v7 -2
[ ~ o , + o/3 + (1 - p)~y + (1 - 0)¢1
dy,
y~U i
and 1
y
K 3 = (n - 1)
n y~v
1
+(n-2)
[(1 - p)ay +/3](n -
2)z "-3 dz dy
i z~U i
f
y
A f nPa(n-1)z"-2dzdy.
y~O i Z=O i
Since
KI+K2+K3=v~ 1 - a
+ ~h
n2] + A~ n +
,X~(n, : - 2)
A(1 n2
we obtain: Ev,(A,p) = v~"
(1-
-
,o
+ - -n+ - ' T n( n -
a+
l-p),
or
Aft Aa Ev,(A, P) = -~- + - - + - ~ ( n n
l-p),
n
owing to
Aa 1 - a + - - = nl 2
n(n -1) - n -2 - + -A n 2
k
n(n -1) n2 - A
1 n
Q.E.D.
p)a
W. Giith, B. Peleg / Mathematical Social Sciences 32 (1996) 1-37
37
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