Essay C: Practically Implementable Auction for a Good with
Countervailing Positive Externalities
Kornpob Bhirombhakdi,
a Ph.D. candidate from Faculty of Economics, Chulalongkorn University, Thailand.
[email protected]
And
Tanapong Potipiti, Ph.D.,
Faculty of Economics, Chulalongkorn University, Thailand.
[email protected]
Revised on 29 November 2012
Abstract
This study theoretically presents a new auction design called "take-or-give auction." The auction solves
the free-rider problem in the case of two symmetric and risk-neutral bidders competing for a good with
countervailing positive externalities. The auction makes efficient allocation. Moreover, the extension of
the auction by addition some rules maximizes the seller's expected revenue.
C.1. Background and Motivation
As found in the previous literature [Jehiel et al. (2000) and Bagwell et al. (2007)], the basic auctions -first- and second-price sealed-bid auctions -- for an object with positive externalities are most likely to
fail for efficient allocation or revenue maximizing since the existence of free-rider problem. This study
proposes two new auctions for a positive-externalities object: one efficiently allocates the object and
the other maximizes expected revenue.
Unlike in the case of normal object which a bidder who does not obtain the object (or non-obtainer)
gains zero payoff, in the case of positive-externalities object he indirectly gains positive payoffs from the
obtainer who directly consumes the good. In this study, I study a special case of positive externality
called "countervailing" positive externality. The countervailing positive externalities is a kind of typedependent-positive externalities which its effects negatively relate with type which represents the
payoffs gained from direct consumption of the object. Precisely, since the higher type the more payoffs
are, the higher type the less positive effects are. For instance, Fig. C-1 (on the right) graphically presents
the utility function on a countervailing-positive-externalities object which a bidder has increasing and
decreasing utility in his type for himself being the obtainer and his opponent being the non-obtainer
respectively; while if nobody obtains the object (or the seller keeps it) he gets zero payoff as status quo.
To compare the differences, Fig. C-1 presents the function on a normal object (on the left) and an object
with non-countervailing positive externalities (in the middle).
utility of player i
utility of player i
i obtains
j obtains
or
no bidder obtains type
of i
utility of player i
i obtains
i obtains
j obtains
j obtains
no bidder obtains
type
of i
no bidder obtains
type
of i
[Fig. C-1 Utility function for a normal object (on the left), an object with non-countervailing positive
externalities (in the middle) and with countervailing positive externalities (on the right)]
This exhibit can be a good example for the countervailing-positive-externalities object. Think of two
adjacent towns -- A and B -- competing for a new airport to be built in one town. Precisely, the
government has a budget to build one airport in town A or B. The town bares no cost of building the
airport but bares the maintenance costs of tourist attractions. Each town has its own number of tourist
attractions which is exogenously given by nature (e.g. by geography of the town). The number of
attractions represents the town's type. In the airport town (the town which obtains the airport), there
will be economic boom which the more attractions the more profit (or payoffs) is gained (since the
average revenue per attraction is higher than the average maintenance cost) -- the obtainer has
increasing payoffs in its type. In the non-airport town, since it is close to the airport town it will get quite
substantial revenue from spillover effects of the economic boom. In other words, for any type of the
non-airport town, the town always gets positive profit. Suppose that the economic boom provide fixed
lump-sum revenue for the non-airport town. The non-airport town gets the highest profit when it has no
attraction. While, given that the average maintenance cost is constant in the number of attractions,
since the average revenue is decreasing in the number, the profit is decreasing as well. Therefore, in this
exhibit the airport shows the countervailing-positive-externalities property. 1
Like the case of public good provision in which the contributor with low marginal benefits from his
contribution -- or low type -- avoids contributing in the provision since he wants to do the free ride on
others' contributions, in any basic auction the bidder with low type avoids participating to compete for
the object with positive externalities since he wants to do the free ride on the obtainer's consumption.
The main cause of such the failure is that, the basic auction sets the rules which the highest-bid bidder
pays and obtains the object hence it does not provide proper incentives for low-type bidders who does
not want to pay nor obtain it. Therefore, a new type of auction which solves the problem must be
implemented.
This article consists of: section C.2. reviews related issues from previous literature; C.3. applies a case of
two discrete types under perfect information to present the failure of basic auction (in C.3.1.) and the
process of the new proposed auction including its various revenue-enhancing extensions (in C.3.2. and
C.3.3.); section C.4. extends the analysis to the case of two discreet types under imperfect environment;
since there are some issues which should be presented before we deal with a formal analysis of
continuous type, section C.5. extends the analysis to the case of four discreet types to present the
issues; section C.6. formally analyzes the case of continuous type under imperfect information; last,
section C.7. provides some concluding remarks. Proofs are provided in Appendix.
1
We may mathematically express as 𝐶(𝑡) = 𝑐𝑡 (where 𝐶 is the total cost, 𝑐 is the average variable cost per unit of
attractions and 𝑡 is type, or the number of attractions), 𝑅𝐴 (𝑡; 𝑃) = 𝐴 + 𝑃𝑡 (where 𝑅𝐴 is the total revenue of the
airport town, 𝐴 is the fixed revenue which and 𝑃 is the average revenue per unit of attractions such that 𝑃 > 𝑐)
and 𝑅𝑁𝐴 = 𝐵 (where 𝑅𝑁𝐴 is the total revenue of the non-airport town and 𝐵 is the lump-sum revenue from
economic boom).
C.2. Literature Review
Since each bidder has privately-known type which represents his payoffs from consuming the auctioned
object -- the more type the more payoffs are -- and equivalently represents his willingness to pay for it,
auction is a mechanism in which a seller applies to gather potential bidders with various types for higher
expected revenue (henceforth, the seller's "revenue" always means his expected one) than selling in a
vendor. Rather than the concern of earning high revenue, some sellers (like government) may concern
the efficient allocation of the object which, mostly, is implemented when allocating the object to the
highest-type bidder.
For a case of normal object -- there is no externality -- Myerson (1981) characterized the revenuemaximizing auction with n-bidders for an indivisible object. The study showed under the directmechanism setting (which bidders report their types and the seller specifies each bidder's payment and
how to allocates the object). However, the direct-mechanism auction is too abstract and hard to be
practically implemented, hence the study also showed a practically implemented version of the
revenue-maximized auction (which is the second-price sealed-bid auction with reservation price).
After the study of Myerson (1981), the interests have been turned into the cases of auction for an object
with specific properties. Related to this study, the property is countervailing positive externalities. In
section C.2.1., it reviews the differences in modeling externalities. Moreover, in this study the auctions
which are designed for efficient allocation and revenue maximization in the case of an object with
countervailing positive externalities are related to countervailing incentives, hence section C.2.2.
reviews this issue. Last, in section C.2.3., I discuss about the optimal rules for revenue maximization in
an auction for an object with externalities.
C.2.1. Externalities
By how are the payoffs of the bidders who do not obtain the object, or the non-obtainers, we classify
the external effects to: negative (when the payoffs are negative) and positive (when the payoffs are
positive). Precisely, Table C-1 presents payoffs of each player -- i and j -- in the corresponding ex-post
outcome -- i obtains, j obtains or no bidder obtains. For the case of negative-externalities object, the
obtainer gets positive payoffs (= 10) but the non-obtainer gets negative payoffs (= -10). For the case of
positive-externalities object, while the obtainer still gets positive payoffs, the non-obtainer also gets
positive payoffs (= 5). And in either cases, if no bidder obtains the object, both players get zero payoff as
status quo.
[Table C-1 Illustration of payoffs for negative-externalities and positive-externalities objects]
Negative externality
Positive externality
Ex-post outcome Payoffs of player i Payoffs of player j Payoffs of player i Payoffs of player j
i obtains
10
-10
10
5
j obtains
-10
10
5
10
no bidder obtains
0
0
0
0
utility of player i
utility of player i
i obtains
no bidder obtains
j obtains
i obtains
type
of i
j obtains
no bidder obtains
type
of i
[Fig. C-2 Utility function for an objects with constant externalities: negative (on the left) and positive (on
the right)]
Like in Jehiel et al. (1996), the external effects can be constant. Graphically, Fig. C-2 presents the
constant externalities: negative on the left and positive on the right. However, the constant externalities
is not likely to be happened in reality. Like competing for a new cost-reduction technology which a firm's
type is its shared profit in the market, when compared to a small shared-profit firm a big shared-profit
firm not only gets higher profit from being the obtainer but also puts more negative externalities on the
non-obtainers [Jehiel et al. (2000)].
Hence, later studies modeled the external effects to be inconstant which depend on types of both
obtainer's and non-obtainers' (which is called type-dependent externalities) or on the identity of the
obtainer (which is called identity-dependent externalities).
Precisely, like in Jehiel et al. (1996), the identity-dependent externalities is determined by who is the
obtainer. For instance, Table C-2 numerically presents the payoffs in the case of negative-externalities
with identity-dependent object. Notice that the external effects on the player i depend on who is the
obtainer. Also, Fig. C-3 draws the utility function in the case of constant-negative externalities and
identity dependent where the player j causes more effects than the player k.
[Table C-2 Illustration of payoffs for negative-externalities with identity-dependent object]
Ex-post outcome Payoffs of player i
i obtains
10
j obtains
-10
k obtains
-5
no bidder obtains
0
utility of player i
i obtains
no bidder obtains
k obtains
type
of i
j obtains
[Fig. C-3 Utility function for an object with identity dependent and constant-negative externalities]
Like in Jehiel et al. (2000), the type-dependent externalities is determined by the bidders' types. As
presented in the Table C-3 for the case of negative externalities, besides the ex-post outcome, the
player's type and his opponent's type also affects the external effects; hence affect the payoffs.
Precisely, from the table, if the player i has low type and the player j obtains the object, the player i gets
-10 payoffs if the player j has low type or gets -20 payoffs if the player j has high type -- the effects
depend on his opponent's type. Also, if the player j obtains the object and has low type, the player i gets
-10 payoffs if he has low type or gets -5 payoffs if he has high type -- the effects depends on his type. Fig.
C-4 presents the case of type-dependent negative externalities.
[Table C-3 Illustration of payoffs for negative-externalities with type-dependent object]
Ex-post outcome
i obtains
j with low type
obtains
j with high type
obtains
no bidder obtains
Payoffs of player i Payoffs of player i
with low type
with high type
10
15
-10
-5
-20
-15
0
0
utility of player i
i obtains
no bidder obtains
type
of i
j with low type obtains
j with high type obtains
[Fig. C-4 Utility function for an object with type dependent and negative externalities]
In this study, like in Chen et al. (2010) we focus on the decreasing-type-dependent positive externalities
which is called "countervailing" positive externalities (see Fig. C-1). As discussed in the study, this case
brought more difficulties in characterizing the revenue-maximizing auction since it dealt with
countervailing incentives (which will be explained in the next section); hence it is more interesting than
other cases.
C.2.2. Countervailing incentives
Lewis et al. (1989) which studied the optimal mechanism in a principle-agent problem related to firm's
production precisely explained that "countervailing incentives exist when the agent has an incentive to
understate his private information for some of its realizations, and to overstate it for others." And as a
consequence, "the agent's rents generally increase with the realization of his private information over
some ranges, and decrease over other ranges." Like in Brocas (2007) for the case of negative
externalities and Chen et al. (2010) for the case of positive externalities, the same things also happened
in the auction settings with countervailing incentives.
Brocas (2007) characterized the revenue-maximizing auction for an indivisible object with typedependent-negative externalities; and the auction had two bidders who suffered from the external
effects symmetrically (hence the effects are identity independent). The study showed three cases which
varied the relationship between the external effects and type: two cases (strongly and weakly) with
decreasing relationship and one with increasing relationship.
Among the cases, the weakly-decreasing case presented the countervailing incentives in the optimal
mechanism. Here, I conclude some (technically) interesting characteristics which are related to this
study:
i) Some specific optimal rules were necessary in the revenue-maximizing auction.
ii) The binding type (or the critical type) which a bidder who has this type gets the expected
payoffs as his reservation payoffs from the designed mechanism was not the highest nor
lowest type but the interior type; which if compare to the case of normal object, the
binding type is the lowest one. Moreover, around the binding type there were some
types which were pooled together. In other words, the optimal mechanism did not
induce the separated equilibrium for the whole-type range. Also, like the finding in
Lewis et al. (1989), a bidder with a type which was outside the pooled region was
induced by providing non-monotonic-information rents (or the countervailing
incentives).
iii) Like Myerson (1981), besides the characterizations of the revenue-maximizing auction in
direct-mechanism setting, the study presented a practically implementable version. In
Brocas (2007), the second-price sealed-bid auction with proper optimal rules was
practically implementable as the revenue-maximizing auction.
In Chen et al. (2010) which characterized the revenue-maximizing auction in the case of countervailing
positive externalities, the study found a countervailing-incentives auction as its revenue-maximizing
mechanism. The study similarly dealt with selecting the binding type which was interior, constructing
the pooling equilibrium in the middle region and providing countervailing incentives in other regions.
But, since the study presented only the optimal auction in direct-mechanism setting, it is still
questionable whether what auction should be the practically implementable version and what kind of
rules is necessary in the optimal auction.
In this study, it fulfills the missing part in Chen et al. (2010) which to present the practically
implementable auction (and hopefully that we can get the auction which is equivalent to the revenuemaximizing one).
C.2.3. Optimal rules for a revenue-maximizing auction
As mentioned, some proper-optimal rules are necessary to design a revenue-maximizing auction in the
case of object with externalities. Besides Brocas (2007), Jehiel et al. (1996) also supported that the rules
are necessary.
Precisely, in the revenue-maximizing auction, Jehiel et al. (1996) discussed that optimal rules (or called
"threats" in the study) will "leave a nonparticipating buyer with the lowest possible level of utility." For
instance, as presented in Fig. C-1 for the case of normal object, the lowest possible level of utility
happens when a bidder is a non-obtainer; hence in the second-price sealed-bid auction with reservation
price which is the revenue-maximizing auction for a normal object the rule is the commitment in which
the object is always allocated to some participating bidders [Myerson (1981)]; and always leave any
nonparticipating bidder with the lowest utility (or zero).
Similarly, for the case of negative-externalities object, the lowest utility happens when a bidder is
suffered from the negative effects -- when he is a non-obtainer. Like in Jehiel et al. (1996) and Brocas
(2007), the seller commits to allocate the object to any participating bidders and leaves the negative
effects to a nonparticipating bidder.
For the case of positive-externalities object, as presented in the Fig. C-1, the lowest utility happens when
the seller keeps the object. The optimal rule is most likely to commit no sale at all if somebody avoids
participating. However, there has been no study explored the revenue-maximizing auction for the object
by implementing the rule; even in Chen et al. (2010), since it characterized the optimal auction under
the direct-mechanism setting, it did not relate its mechanism with any rule.
Besides, there are two important rules which may be concerned as the optimal rules: no coalition
between bidders and no re-selling. Precisely, both rules are applied to guarantee competitive
environment in the auction which directly enhances the seller's revenue. Practically, the study implicitly
assumed that money transfer among bidders is not feasible, hence neither collusion nor re-selling can
happen [Jehiel et al. (1996)].
There were some arguments about implementing those optimal rules. Like in Jehiel et al. (2000), the
study argued that any rule which make "the seller can extract payments" from any non-obtainer need
"an unrealistically strong commitment power" from the seller. Hence to avoid implementing those
unrealistic rules, the study applied the entry fee and reservation price as the realistic rules in auction.
In my opinion, any auction rule is realistic and practically implementable as long as the nonparticipating
bidders pay no price (while the Jehiel et al. (2000) defined by no price for non-obtainers). Since if
participating bidders accept and commit to the rules of auction, the also commit to pay even they are
non-obtainers. Hence, under my opinion, all of the rules which have been discussed here are realistic
and practically implementable.
C.3. Simple illustration: two discrete types under perfect information
In this section, to ease your understanding before dealing with continuous setting which is more
complicated, I simply apply the case of two symmetric bidders and discrete types -- high type and low
type as presented in Table C-5. This section composes of three sub-sections which present each
important point. The section C.3.1. presents the free-rider problem in the second-price sealed-bid
auction. In the section C.3.2., the "take-or-give auction" is presented to show how it is processed and
fixes the problem. Last, I further explain how to increase the seller's revenue by adding more rules in the
take-or-give auction.
[Table C-5 Payoffs for the two-discrete-type illustration]
Payoffs when being the obtainer Payoffs when being the non-obtainer
No obtainer
𝑊
𝐿
High type
8$
4$
0$
Low type
4$
8$
0$
Type
C.3.1. Free-rider problem in the second-price sealed-bid auction
This section presents how the basic auction (like second-price sealed-bid auction) is most likely to fail
being the optimal auction -- either efficient or revenue-maximizing -- for any object with positive
externalities; since the free-rider problem, especially, exists if a bidder's payoffs when being the nonobtainer 𝐿 are higher than when being the obtainer 𝑊. Precisely, as presented in the Table C-5, a bidder
with low type gets higher payoffs when he is the non-obtainer, 𝐿(𝑙𝑜𝑤 𝑡𝑦𝑝𝑒) = 8 > 4 = 𝑊(𝑙𝑜𝑤 𝑡𝑦𝑝𝑒);
since the low-type bidder will try to avoid being the obtainer, he is the free rider.
Bagwell et al. (2007) and Jehiel et al. (2000) presented the findings which showed the free-rider problem
in the first- and second-priced sealed-bid auction for a positive-externalities object respectively. Here, I
aim to make a simple analysis to show the problem; we apply a two-discrete-type illustration, as
presented in the Table C-5, in the second-price sealed-bid auction under the perfect-information
environment (the problem also extends to the case of imperfect information and continuous type); and
the auction has two potential symmetric bidders.
[Table C-6 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue of each possible type realization in the second-price sealed-bid auction]
Type
realization
High and
High
Equilibrium strategy
Ex-ante outcome
(payment and allocation)
Expected
bidder'
surplus
Both bid 4$.
Each bidder has 0.5 chance
of being the obtainer. The
obtainer pays 4$, and no
payment for the nonobtainer.
4$
Seller's
revenue
(from
both
bidders)
4$
Low and
Low
Mixed-equilibrium strategy
between not participating (with
p chance) and participating with
zero bid (with 1-p chance).2
May or may not have the
obtainer. In either case,
bidders pay no price.
6∗
(1 − 𝑝2 )$
0$
As presented in the Table C-6, according to symmetry, the situation has two possibilities of type
realization -- high-high and low-low. If both have high type, they bid 4$ which is the wiliness to pay
comparing between alternatives -- payoffs of being the obtainer and non-obtainer (bid = 𝑊 − 𝐿). To
see this, let both bidders play higher bid strategy, say, 5$. The expectation on bidder's surplus is 0.5*(8 5) + 0.5*4 = 3.5$. The 5$-strategy is not in equilibrium since a bidder would like to deviate by submitting
a little lower bid hence he certainly is the non-obtainer and gets 4$ surplus. Similarly, let both bidders
play lower bid strategy -- 3$-strategy. The surplus is 4.5$ which is not in equilibrium since deviation by
submitting a little higher bid certainly makes him be the obtainer and gets 8 - 3 = 5$ surplus. Iteratively,
bidding 4$ is in equilibrium.
For the case of low-low realization, the precise calculation is provided in the footnote and here I roughly
explain how the mixed strategy is in equilibrium. First, notice that a low-type bidder does not want to be
the obtainer and does want to prevent no sale event where the seller keeps the object and no bidder
obtains it. From these incentives, if a low-type bidder certainly knows that his opponent will participate,
he will not participate; or if he knows that the opponent will not participate, he will participate with the
lowest possible bid (which is 0$ bid in this case). Since he does not know what his opponent will play, he
randomizes the strategies.
As mentioned that the free-rider problem happens if some bidder has low type which 𝐿 > 𝑊, in the
low-low realization the problem happens since the low-type bidder would like to be the non-obtainer
and do the free ride on his opponent's consumption. This also implies that the problem always exists in
any basic auction in which the highest-bid bidder is the obtainer. Hence, it is necessary to design a new
auction which provides proper incentives for the bidders such as allowing the highest-bid bidder select
for himself whether to be the obtainer or not.
C.3.2. Illustration of take-or-give auction with second-price payment
As presented in the previous section, any basic auction always fails to fix the free-rider problem since it
does not provide the proper incentives for bidders, especially for the low-type one. Hence, this study
2
For the case of low-low realization, to find the p chance, consider this payoff matrix
P
N
Participating with zero bid
6,6 4,8
(P)
Not participating
8,4 0,0
(N)
2
we will get 𝑝 = . (The utility from the strategy (P,P) is 6$ for each player since the seller randomly allocates the
3
object with 0.5 chance).
proposes a new auction which can fix the problem by providing the proper incentives -- allowing the
highest-bid bidder select for himself whether to take the object and be the obtainer or to give the object
to his opponent and be the non-obtainer. In other words, instead of competing to obtain the object, the
new auction lets the bidders compete for their desired allocation of the object. The auction is called
"take-or-give auction with second-price payment."
Precisely, the take-or-give auction allows each bidder submit a doublet which composes of bid and the
demand of allocation of the object. The demand can be either to take and be the obtainer or to give his
opponent the object (for free as a gift) and be a non-obtainer.
The payment is the second-price payment conditioned on their demands. Precisely, if both bidders
submit the same demand, the highest-bid bidder pays his opponent's bid as the second price. Or if both
bidders submit different demands, they pay nothing (as a zero reservation price conditioned on each
demand).
[Table C-7 Illustrations of the payment and allocation mechanisms of the take-or-give auction with
second-price payment]
Scenario
Strategies
Allocation
Payment
Both demand to take.
Bidder i pays 7$.
1
Bidder i bids 10$.
Bidder i is the obtainer.
Bidder j pays nothing.
Bidder j bids 7$.
Both demand to give.
Bidder i is the giver and
Bidder i pays 7$.
2
Bidder i bids 10$.
is the non-obtainer.
Bidder j pays nothing.
Bidder j bids 7$.
Bidder j is the obtainer.
Bidder i bids 10$ and demands to take.
3
Bidder i is the obtainer.
Both pay nothing.
Bidder j bids 7$ and demands to give.
Bidder i is the giver and
Bidder i bids 10$ and demand to give.
4
is the non-obtainer.
Both pay nothing.
Bidder j does not participate.
Bidder j is the obtainer.
The Table C-7 illustrates how the payment and allocation mechanisms of the take-or-give auction with
second-price payment are determined. In the first and second scenarios, since both bidders submit the
same demand the allocation is made according to the highest-bid bidder and he pays the second price;
while the other bidder pays nothing. In the third scenario, since both bidders submit different demands
the allocation is made according to the highest-bid bidder (and virtually the other bidder gets desired
allocation) but both pays nothing. Last, in the fourth scenario where the bidder j does not participate,
the allocation is made according to the highest-bid bidder but he pays nothing.
The take-or-give auction with second-price payment is processed in three steps as follows:
1. Each bidder submits his doublet of bid and demand (which is taking or giving).
2. The seller selects the highest-bid bidder's demand and allocates the object accordingly. If the
bids are tied, the seller prefers to allocate the object to a bidder who demands to take
than to give.
3. The highest-bid bidder pays equal to:
- his opponent's bid if they submit the same demand,
- or zero if his opponent submits different demand or does not participate.
Note that the take-or-give auction has the following important assumption:
Assumption 1
The seller can allocate the object to any nonparticipating bidder.3
[Table C-8 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue of each possible type realization in the take-or-give auction with second-price payment]
Type
realization
Equilibrium
strategy
High and
High
Both bid 4$ and
demand to take.
Low and
Low
Both bid 4$ and
demand to give.
Ex-ante outcome
(payment and allocation)
Each bidder has 0.5 chance of being the
obtainer. The obtainer pays 4$, and no
payment for the non-obtainer.
Each bidder has 0.5 probability of being
the giver. The giver pays 4$, and gives
his opponent the object without
payment.
Expected
bidder's
surplus
Seller's
revenue
(from both
bidders)
4$
4$
4$
4$
According to the payoffs defined in Table C-5, the Table C-8 presents the symmetric-equilibrium
strategy, ex-ante outcome, expectation on bidder's surplus and seller's revenue in each type realization
under the take-or-give auction with second-price payment and perfect-information environment. To
show that the strategies in the table are in equilibrium, we consider the high-high realization; if both
submit demand to give, this is not equilibrium since a bidder would like to deviate by demanding to
take; if both submit higher bid (say 5$) with demand to take, this is not equilibrium since a bidder would
like to deviate by submitting lower bid; and if both submits lower bid (say 3$) with demand to take, this
is not equilibrium since a bidder would like to deviate by submitting higher bid. Iteratively, submitting 4$
bid with demand to take is in equilibrium. Similar arguments can be done in the case of low-low
realization. Hence, the equilibrium strategies are as specified in the table.
3
As discussed in Jehiel et al. (1996) that in a feasible auction the seller could not allocate the object to a
nonparticipating bidder, the take-or-give auction imposes the different assumption. To claim for its credibility, let
us consider the following two explanations. First, allocating the object to a nonparticipating bidder for free is
similar as giving a gift. Second, like in a business meeting which all members should participate to vote and make
decisions on their own wills, the participating members lose their time and efforts (as the payment) but gain the
chance to convince the final decisions as they want; while a nonparticipating member loses no time nor effort in
the meeting (as without payment) but also loses the chance of making the decisions and totally agrees with the
final decisions from the participating members.
Another questionable issue is that, a nonparticipating bidder avoids participating since he does not want
to obtain and consume the object, will he consume it if he obtains it as a gift? Logically, he will; since not
consuming leaves him zero payoff while consuming yields him 𝑊 > 0.
Notice the change in the equilibrium strategy of the low-low realization. Compare to the basic auction,
the take-or-give auction induces the low-type bidders to participate and compete for the desired
allocation; hence the free-rider problem is totally fixed.
Precisely, we observe: i) the auction leaves no incentive for the low-type bidder to avoid participating;
hence the free-rider problem is totally fixed and ii) The seller earns higher revenue.
C.3.3. Illustration of revenue-enhancing rules
Next, we consider how the additional rules can enhance the level of revenue. In this study, we apply two
rules: entry fees conditioned on the demand (which the fee of participating with demand to take can be
different from the fee of participating with demand to give) and no sale condition (which the seller
commits no sale at all if any potential bidder does not participate). The following sections present three
important issues: how the take-or-give auction with the rules is processed, how the rules affect behavior
and how the rules affect revenue and efficiency of allocation. In the first section, we introduce only the
entry fees. Then, we introduce both fees and no sale condition.
- The take-or-give auction with second-price payment and entry fees: As mentioned that the
fees are differentiated according to the submitted demand. Precisely, the seller designs a menu of entry
fees (𝐸𝑔𝑖𝑣𝑒 , 𝐸𝑡𝑎𝑘𝑒 ) which a bidder pays 𝐸𝑔𝑖𝑣𝑒 if he demands to give; or pays 𝐸𝑡𝑎𝑘𝑒 if he demands to take.
The take-or-give auction with second-price payment and entry fees is processed as follows:
1. Each bidder submits his doublet which composes of bid and entry fee -- 𝐸𝑡𝑎𝑘𝑒 if he demands
to take or 𝐸𝑔𝑖𝑣𝑒 if he demands to give.
2. The seller selects the highest-bid bidder's demand and allocates the object accordingly. If the
bids are tied, the seller prefers to allocate it to a bidder who pays 𝐸𝑡𝑎𝑘𝑒 to 𝐸𝑔𝑖𝑣𝑒 .
3. The highest-bid bidder additionally pays equal to:
- his opponent's bid if they submit the same entry fee,
- or zero if his opponent submits different entry fees or does not participate.
[Table C-9 Illustrations of the payment and allocation mechanisms of the take-or-give auction with
second-price payment and entry fees]
Scenario
Strategies
Allocation
Payment
Both submit 𝐸𝑡𝑎𝑘𝑒 .
Bidder i is the
Both pay 𝐸𝑡𝑎𝑘𝑒 as the fee.
5
Bidder i bids 10$.
obtainer.
Bidder i additionally pays 7$.
Bidder j bids 7$.
Bidder i is the giver
Both submit 𝐸𝑔𝑖𝑣𝑒 .
and
Both pay 𝐸𝑔𝑖𝑣𝑒 as the fee.
6
is the nonBidder i bids 10$.
Bidder i additionally pays 7$.
obtainer.
Bidder j bids 7$.
Bidder j is the
obtainer.
7
8
Bidder i submits 10$ bid
and 𝐸𝑡𝑎𝑘𝑒 .
Bidder j submits 7$ bid and
𝐸𝑔𝑖𝑣𝑒 .
Bidder i is the
obtainer.
Bidder i pays 𝐸𝑡𝑎𝑘𝑒 and bidder j pays
𝐸𝑔𝑖𝑣𝑒 as the fee.
Bidder i submits 10$ bid
and 𝐸𝑔𝑖𝑣𝑒 .
Bidder j does not
participate.
Bidder i is the giver
and
is the nonobtainer.
Bidder j is the
obtainer.
Bidder i pays 𝐸𝑔𝑖𝑣𝑒 as the fee.
Bidder j pays nothing.
To illustrate the allocation and payment mechanisms, Table C-9 presents 5th-8th scenarios. Compare to
the auction without the fees, a participating bidder pays the entry fee which is corresponding to his
demand and additionally pays the second price if the case is applicable. Precisely, in the 5th scenario,
both submits the same entry fee (which equivalently means that both bidders submit demand to take),
the highest-bid bidder pays 𝐸𝑡𝑎𝑘𝑒 plus second price and the allocation is made according to his demand;
while the other pays only the fee 𝐸𝑡𝑎𝑘𝑒 . Similar to the 5th scenario, the 6th scenario illustrates the case
which both bidders pay 𝐸𝑔𝑖𝑣𝑒 . In the 7th scenario, both submit different entry fees, each bidder pays the
fee and does not pay the second price; the allocation is made according to the highest-bid bidder's
demand. In the 8th scenario, the other bidder does not participate, the participating bidder pays only the
fee and the allocation is made according to his demand.
[Table C-10 Payoff matrix for high-high type realization in the take-or-give auction with second-price
payment and entry fees when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 𝐸]
High-high realization
Participate with
demand to take
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
1
6 − 𝐸 − 𝑏𝑡𝑎𝑘𝑒
2
1
6 − 𝐸 − 𝑏𝑡𝑎𝑘𝑒
2
Participate with
demand to give
(P,G, 𝑏𝑔𝑖𝑣𝑒 )
4−𝐸
8−𝐸
Not participate
(N)
4
8−𝐸
(P,G, 𝑏𝑔𝑖𝑣𝑒 )
(N)
8−𝐸
4−𝐸
8−𝐸
4
1
6 − 𝐸 − 𝑏𝑔𝑖𝑣𝑒
2
1
6 − 𝐸 − 𝑏𝑔𝑖𝑣𝑒
2
8
4−𝐸
4−𝐸
8
0
0
Next, we will see how the entry-fees rule affects behavior. Similar to the entry-fees rule in the basic
auction, in the take-or-give auction the rule affects behavior by the exclusion effect -- high enough entry
fees may affect some low-willingness-to-pay bidders do not participate. To see the effect, consider the
payoff matrix (as presented in Table C-10) which is derived from the utility function as specified in the
Table C-5. Precisely, each bidder has three strategies: participating with bid 𝑏𝑡𝑎𝑘𝑒 and demand to take
(P,T, 𝑏𝑡𝑎𝑘𝑒 ), participating with bid 𝑏𝑔𝑖𝑣𝑒 with demand to give (P,G, 𝑏𝑔𝑖𝑣𝑒 ) and not participating (N).
Suppose the menu (𝐸𝑔𝑖𝑣𝑒 , 𝐸𝑡𝑎𝑘𝑒 ) = (𝐸, 𝐸). Hence, in the case of high-high realization, under the takeor-give auction with second-price payment and entry fees we can derive the payoff matrix. For instance,
if both play (P,T, 𝑏𝑡𝑎𝑘𝑒 ), the seller randomly allocates the object with 0.5 chance hence each bidder has
1
the expected surplus (or payoffs) as −𝐸 + 0.5 ∗ (8 − 𝑏𝑡𝑎𝑘𝑒 ) + 0.5 ∗ 4 = 6 − 𝐸 − 2 𝑏𝑡𝑎𝑘𝑒 .
From the payoff matrix, we can see that, given any mixed strategy of the second bidder, increasing 𝐸
does not affect the expected surplus of the first bidder if he plays (N) but directly decreases the
expected surplus of the first bidder if he plays (P,T, 𝑏𝑡𝑎𝑘𝑒 ) or (P,G, 𝑏𝑔𝑖𝑣𝑒 ); hence we see the exclusion
effect. Precisely, high enough level of 𝐸 makes the expected surplus from playing (N) be higher than
plying (P,T, 𝑏𝑡𝑎𝑘𝑒 ) or (P,G, 𝑏𝑔𝑖𝑣𝑒 ).
By designing a proper menu (𝐸𝑔𝑖𝑣𝑒 , 𝐸𝑡𝑎𝑘𝑒 ), the rule also increases revenue. To illustrate, consider the
case which 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 4. The payoff matrix in Table C-10 can be derived as presented the Table C11.
[Table C-11 Payoff matrix for high-high type realization in the take-or-give auction with second-price
payment and entry fees when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 4]
High-high realization
Participate with
demand to take
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
1
2 − 𝑏𝑡𝑎𝑘𝑒
2
1
2 − 𝑏𝑡𝑎𝑘𝑒
2
Participate with
demand to give
(P,G, 𝑏𝑔𝑖𝑣𝑒 )
0
4
Not participate
(N)
4
4
(P,G, 𝑏𝑔𝑖𝑣𝑒 ) (N)
4
0
4
4
1
2 − 𝑏𝑔𝑖𝑣𝑒
2
1
2 − 𝑏𝑔𝑖𝑣𝑒
2
8
0
0
8
0
0
[Table C-12 Payoff matrix for high-high type realization in the take-or-give auction with second-price
payment and entry fees when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 4 and (P,G, 𝑏𝑔𝑖𝑣𝑒 ) is eliminated]
High-high realization
Participate with
demand to take
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
Not participate
(N)
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
1
2 − 𝑏𝑡𝑎𝑘𝑒
2
1
2 − 𝑏𝑡𝑎𝑘𝑒
2
4
4
(N)
4
4
0
0
From the Table C-11, since (P,G, 𝑏𝑔𝑖𝑣𝑒 ) is strictly dominated we can eliminate the strategy and derive
Table C-12. Then, consider the bidder's problem:
1
max 𝐸𝑢 = 4𝑞 − 𝑝 [6𝑞 + 𝑞𝑏𝑡𝑎𝑘𝑒 − 4]
𝑝,𝑏𝑡𝑎𝑘𝑒
2
where 𝑝 and 𝑞 are the probability that he and his opponent play (P,T, 𝑏𝑡𝑎𝑘𝑒 ) respectively. Optimally, the
2
symmetric-equilibrium strategy is (𝑝∗ , 𝑏𝑡𝑎𝑘𝑒 ∗ ) = (3 , 0). Hence, the revenue is
16
3
which is higher than
the revenue in the take-or-give auction without the entry fees (see Table C-8, the revenue is 4$).
Moreover, since the exclusion effect the auction makes inefficient allocation. Precisely, Table C-13
presents the symmetric-equilibrium strategy, ex-ante outcome, expected bidder's surplus and seller's
revenue in the take-or-give auction with second-price payment and entry fees when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 4.
[Table C-13 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue of each possible type realization in the take-or-give auction with second-price payment and
entry fees when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 4]
Type
realization
High and
High
Low and
Low
Equilibrium strategy
Mixed strategy between
participating with 0$ bid and
2
demand to take at 3 chance
1
and not participating at
3
chance.
Mixed strategy between
participating with 0$ bid and
2
demand to give at 3 chance
1
and not participating at 3
chance.
Ex-ante outcome
(payment and allocation)
Expected
bidder's
surplus
Seller's
revenue
(from
both
bidders)
May or may not have the
obtainer. Participating bidder
pays 4$ as fee. There is no
second-price payment.
8
$
3
16
$
3
May or may not have the
obtainer. Participating bidder
pays 4$ as fee. There is no
second-price payment.
8
$
3
16
$
3
- The take-or-give auction with second-price payment and no sale condition: In addition to the
auction with entry fees, we add the no sale condition to the auction. The condition commits no sale at
all if any potential bidder does not participate; in other words, the auction is cancelled. Notice that since
the rules charge no payment to a nonparticipating bidder, as discussed previously, the rules are credible.
The auction process is as follows:
1. Each bidder submits his doublet which composes of bid and entry fee -- 𝐸𝑡𝑎𝑘𝑒 if he demands
to take or 𝐸𝑔𝑖𝑣𝑒 if he demands to give. Not participating is also available which if any
bidder does so both pay nothing and the auction is cancelled.
2. If the auction is not cancelled, the seller selects the highest-bid bidder's demand and allocates
the object accordingly. If the bids are tied, the seller prefers to allocate it to a bidder
who pays 𝐸𝑡𝑎𝑘𝑒 than 𝐸𝑔𝑖𝑣𝑒 .
3. The highest-bid bidder additionally pays equal to:
- his opponent's bid if they submit the same entry fee,
- or zero if his opponent submits different entry fees.
[Table C-14 Illustrations of the payment and allocation mechanisms of the take-or-give auction with
second-price payment, entry fees and no sale condition]
Scenario
Strategies
Allocation
Payment
Bidder i submits 10$ bid and
The seller keeps the object; no bidder
Both pay
𝐸𝑔𝑖𝑣𝑒 .
9
obtains.
nothing.
Bidder j does not participate.
To illustrate the allocation and payment mechanisms of the auction, unlike the 8th scenario in the Table
C-9, Table C-14 presents 9th scenario where bidder j does not participate; hence the auction is canceled,
both bidders pay nothing and the seller keeps the object. For other scenarios where all bidders
participate, the payment and allocation are the same as presented in 5th-7th scenarios in Table C-9.
[Table C-15 Payoff matrix for high-high type realization in the take-or-give auction with second-price
payment, entry fees and no sale condition when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 𝐸]
High-high realization
Participate with
demand to take
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
(P,T, 𝑏𝑡𝑎𝑘𝑒 )
1
6 − 𝐸 − 𝑏𝑡𝑎𝑘𝑒
2
1
6 − 𝐸 − 𝑏𝑡𝑎𝑘𝑒
2
Participate with
demand to give
(P,G, 𝑏𝑔𝑖𝑣𝑒 )
4−𝐸
8−𝐸
Not participate
(N)
0
0
(P,G, 𝑏𝑔𝑖𝑣𝑒 )
(N)
8−𝐸
4−𝐸
0
0
1
6 − 𝐸 − 𝑏𝑔𝑖𝑣𝑒
2
1
6 − 𝐸 − 𝑏𝑔𝑖𝑣𝑒
2
0
0
0
0
0
0
Compare to the auction with entry fees and without the condition, the condition directly affects both
participating and nonparticipating bidders which, as a consequence, both revenue and efficiency of
allocation are increased (recall that the auction with entry fees and without the condition makes
inefficient allocation since the exclusion effect). To illustrate the effects, similar to the Table C-10, under
the take-or-give auction with second-price payment, entry fees and no sale condition the Table C-15
presents the payoff matrix which is derived from the payoffs as specified in the Table C-5. Notice the
difference between the Table C-15 and C-10, under the auction with no sale condition if any potential
bidder does not participate, the auction will be canceled; hence no bidder obtains the object and both
bidders get zero payoff.
Suppose 𝐸𝑔𝑖𝑣𝑒 = 𝐸𝑡𝑎𝑘𝑒 = 4. According to the payoff matrix in Table C-15, the symmetric-equilibrium
strategy is to play (P,T, 𝑏𝑡𝑎𝑘𝑒 ∗) with 𝑏𝑡𝑎𝑘𝑒 ∗ = 4 and any 𝑏𝑔𝑖𝑣𝑒 ∈ ℝ. To see that the strategy constitutes
the equilibrium, directly check that there is no incentive to deviate from the equilibrium strategy. Table
C-16 presents the symmetric-equilibrium strategy, ex-ante outcome, expected bidder's surplus and
seller's revenue in the take-or-give auction with second-price payment, entry fees and no sale condition
when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 4.
[Table C-16 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue of each possible type realization in the take-or-give auction with second-price payment, entry
fees and no sale condition when 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 = 4]
Type
realization
Equilibrium strategy
High and
High
Both participate with 4$
bid and demand to take.
Low and
Low
Both participate with 4$
bid and demand to give.
Ex-ante outcome
(payment and allocation)
Both pay 4$ as the fee.
Each bidder has 0.5 chance of
being the obtainer.
The obtainer additionally pays
4$.
Both pay 4$ as the fee.
Each bidder has 0.5 chance of
being the giver.
The giver additionally pays 4$
and gives the other bidder the
object.
Expected
bidder's
surplus
Seller's
revenue
(from both
bidders)
0$
12$
0$
12$
As presented in the Table C-16, since at the equilibrium there is no chance that the seller keeps the
object, unlike the result in the auction with fees but without condition, the take-or-give auction with
second-price payment, entry fees and no sale condition increases the efficiency of allocation. Moreover,
since all bidder's surplus is extracted, the auction maximizes revenue.
C.4. Imperfect-information and two-discrete-type illustration
In this section, we extend the analysis to two-discrete types under imperfect information with two riskneutral and symmetric bidders. Like the case of perfect information, the payoffs are as specified in Table
C-5. For simplicity, let the prior probability of having high type be 0.5.
The analysis will draw the parallel between perfect- and imperfect-information environments by
identifying the free-rider problem in the basic auction, fixing the problem by the take-or-give auction
and enhancing revenue by entry-fee and no-sale-condition rules.
C.4.1. Free-rider problem (imperfect information)
Similar to the case of perfect information, according to the payoffs as specified in Table C-5, the free
rider is a low-type bidder whose incentives are to avoid participating in the auction and also to prevent
the no sale event by participating with zero bid. To identify the problem under the imperfect
information, in the second-price sealed-bid auction the expected utility function of a low-type bidder is:
1
𝐸𝑢𝑖 (𝑙𝑜𝑤 𝑡𝑦𝑝𝑒, 𝑏𝑖 |𝑏𝑗 ) = 𝑝𝑟(𝑏𝑗 < 𝑏𝑖 )(4 − 𝑏𝑗 ) + 𝑝𝑟(𝑏𝑗 = 𝑏𝑖 ) (6 − 𝑏𝑗 ) + 𝑝𝑟(𝑏𝑗 > 𝑏𝑖 )(8)
2
where 𝑏𝑖 is the bidder's bid; 𝑏𝑗 is his opponent's bid; 𝑝𝑟(∙) is the probability of each event. The firstorder condition with respect to 𝑏𝑖 of the expected utility shows that the expected utility is decreasing in
𝑏𝑖 ; hence the low-type bidder always submits the lowest possible bid if he participates in the auction.
The finding presents the free-rider problem in the basic auction.
C.4.2. The take-or-give auction with second-price payment (imperfect information)
Similar to the illustration in the case of perfect information, under the take-or-give auction with secondprice payment the symmetric-equilibrium strategy is as presented in the Table C-17.
[Table C-17 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue under imperfect information in the take-or-give auction with second-price payment]
Type
realization
Equilibrium strategy
High
Participate with 4$
bid and demand to
take.
Low
Participate with 4$
bid and demand to
give.
Ex-ante outcome
(payment and allocation)
Expected
bidder's
surplus
Seller's
revenue
(from both
bidders)
If both submit demand to take, each
bidder has 0.5 chance of being the
obtainer; the obtainer pays 4$.
If both submit demand to give, each
bidder has 0.5 chance of being the
giver; the giver pays 4$.
If both submit different demand,
both pay nothing and the high-type
6$
2$
bidder obtains the object.
To see that the strategy as specified in the Table C-17 is in (Bayes-Nash) equilibrium, consider first that a
low-type bidder who has 𝐿 > 𝑊 always submits demand to give and a high-type bidder who has 𝑊 > 𝐿
always submits demand to take. To show this, we derive the expected utility of a low-type bidder as
𝐸𝑢𝑖 (𝑙𝑜𝑤 𝑡𝑦𝑝𝑒, 𝑑𝑖 , 𝑏𝑖 |𝑑𝑗 , 𝑏𝑗 )
= 𝑝𝑟(𝑏𝑗 < 𝑏𝑖 ){8 − 4𝑑𝑖 − 𝑝𝑟(𝑑𝑖 = 𝑑𝑗 )𝑏𝑗 }
1
+ 𝑝𝑟(𝑏𝑗 = 𝑏𝑖 ) {𝑝𝑟(𝑑𝑖 = 𝑑𝑗 ) (4 − 2𝑑𝑖 − 𝑏𝑗 )} + 𝑝𝑟(𝑏𝑗 > 𝑏𝑖 ){4 + 4𝑑𝑗 }
2
where for the bidder i 𝑑𝑖 = 1 for taking or 𝑑𝑖 = 0 for giving. From the first-order condition, regardless of
𝑏𝑖 the expected utility function is decreasing in 𝑑𝑖 ; hence it is optimal for a low-type bidder to submit
demand to give. Similarly, we can derive the expected utility function of a high-type bidder to show his
optimal demand is to take. Last, to show that bidding 4$ constitutes the equilibrium, the similar
arguments as presented in section C.3.1. can be applied here.
As presented in the Table C-17, similar to what we have seen in the case of perfect information, the
take-or-give auction can fix the free-rider problem. Moreover, the ex-ante outcome shows that the
auction efficiently allocates the object. (Later when we formally analyze the case of continuous type the
result of efficient allocation of the auction is proposed in the Proposition 3).
C.4.3. Revenue-enhancing rules: entry fees and no sale condition (imperfect information)
[Table C-18 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue under imperfect information in the take-or-give auction with second-price payment, entry fees
and no sale condition]
Type
realization
Equilibrium strategy
High
Participate with 4$
bid and 𝐸𝑡𝑎𝑘𝑒 (or
demand to take).
Low
Participate with 4$
bid and 𝐸𝑔𝑖𝑣𝑒 (or
demand to give).
Ex-ante outcome
(payment and allocation)
Each bidder pays the submitted fee
-- 𝐸𝑡𝑎𝑘𝑒 for high-type bidder and
𝐸𝑔𝑖𝑣𝑒 for low-type one.
If both submit 𝐸𝑡𝑎𝑘𝑒 , each bidder
has 0.5 chance of being the
obtainer; the obtainer additionally
pays 4$.
If both submit 𝐸𝑔𝑖𝑣𝑒 , each bidder
has 0.5 chance of being the giver;
the giver additionally pays 4$.
If both submit different fees, there
is no additional payment and the
Expected
bidder's
surplus
Seller's
revenue
(from both
bidders)
6 − 𝐸𝑡𝑎𝑘𝑒 $
6 − 𝐸𝑔𝑖𝑣𝑒 $
2+
𝐸𝑡𝑎𝑘𝑒 +
𝐸𝑔𝑖𝑣𝑒 $
high-type bidder obtains the
object.
Similarly, the Table C-18 presents the symmetric-equilibrium strategy, ex-ante outcome, expected
bidder's surplus and seller's revenue in the take-give-auction with second-price payment, entry fees and
no sale condition under the imperfect information and two discrete types (the payoffs of each type are
presented in Table C-5). To proof for the equilibrium strategy, the same arguments as presented in the
previous section C.4.2. can be applied.
Compare to the bidder's surplus in the take-or-give auction without entry fees or no sale condition (see
Table C-17), with the entry fees and no sale condition the auction can extract more surplus by setting
the proper levels of fees. Precisely, the optimal menu is (𝐸𝑡𝑎𝑘𝑒 ∗ , 𝐸𝑔𝑖𝑣𝑒 ∗ ) = (6,6) which can extract all
the bidders' surplus. Hence, under this setting the take-or-give auction with second-price payment,
entry fees and no sale condition maximizes the seller's expected revenue.
However, since there are only two types in the setting, the revenue-maximizing auction extracts all the
surplus; in other words, the seller pays no information rent. This result will not hold when there are
more than two types. In the next section C.5., we will explore the case of four discrete types under
imperfect information.
C.5. Four discrete types under imperfect information: separating equilibrium
for the whole-type range VS middle-pooling equilibrium
[Table C-19 Payoffs for the three discrete types (10 > 𝐴 > 𝐵 > 0 and 𝐴 + 𝐵 = 10)]
No
Payoffs when being the obtainer Payoffs when being the non-obtainer
obtainer
𝑊
𝐿
High type (H)
10$
0$
0$
Middle type 1 (N)
A$
B$
0$
Middle type 2 (M)
B$
A$
0$
Low type (L)
0$
10$
0$
type
This section presents the take-or-give auction with second-price payment, entry fees and no sale
condition under the setting of four discrete types and imperfect information. The payoffs of each type
are presented in the Table C-19.
There are two important issues which are best to be pointed out by this setting. First, the take-or-give
auction provides countervailing incentives -- the information rent is decreasing in types over one range
but increasing in types over other range. Second, which is the main issue in this section, under some
circumstances, designing the incentives to make pooling equilibrium at the middle-type range and leave
other ranges for separating equilibrium yields higher revenue than designing the incentives to make
separating equilibrium for the whole-type range.
To present the issues, this section composes of two parts. Section C.5.1. presents the designing for
separating equilibrium for the whole-type range; this section presents the first issue. Section C.5.2.
presents the designing for pooling equilibrium at middle-type range and presents the second issue.
C.5.1. Separating equilibrium for the whole-type range in the take-or-give auction with
second-price payment, entry fees and no sale condition
As mentioned, under the take-or-give auction with second-price payment, entry fees and no sale
condition this section presents that the auction pays countervailing incentives to induce separating
equilibrium. We will apply the payoffs and types as specified in the Table C-19 under imperfect
information; and there are two symmetric bidders with risk neutrality. Each bidder is randomly and
independently drawn his type from the four possible realizations -- H, N, M and L (ordered from high to
low type). For simplicity, let the prior probability of having each type be 0.25. Also assume 10 > 𝐴 >
𝐵 > 0 and 𝐴 + 𝐵 = 10 (hence 5 < 𝐴 < 10).
Assumption 2
The seller designs the take-or-give auction with second-price payment, entry
fees and no sale condition which all bidders participate.
Under the auction, we are interested in the equilibrium which all bidders participate (Assumption 2). In
designing separating equilibrium for the whole-type range, the seller designs a menu of entry fees
(𝐸𝑔𝑖𝑣𝑒 , 𝐸𝑡𝑎𝑘𝑒 ) which solves the following system:
1
1
1
1
−𝐸𝑡𝑎𝑘𝑒 + (35 − 𝑏𝐻 − 𝑏𝑁 ) ≥ 0, −𝐸𝑡𝑎𝑘𝑒 + (15 + 𝐴 − 𝑏𝑁 ) ≥ 0,
4
2
4
2
1
1
1
1
−𝐸𝑔𝑖𝑣𝑒 + (15 + 𝐴 − 𝑏𝑀 ) ≥ 0, −𝐸𝑔𝑖𝑣𝑒 + (35 − 𝑏𝐿 − 𝑏𝑀 ) ≥ 0,
4
2
4
2
1
1
1
1
1
1
𝑏𝐻 + 𝑏𝑁 ≤ 20, ∆𝐸 + (20 − 𝑏𝐻 − 𝑏𝑁 + 𝑏𝑀 ) ≥ 0, ∆𝐸 + (30 − 𝑏𝐻 − 𝑏𝑁 + 𝑏𝐿 + 𝑏𝑀 ) ≥ 0,
4
2
2
4
2
2
1
1
1
1
1
1
𝑏𝐻 − 𝑏𝑁 ≥ 4𝐴 − 20, ∆𝐸 + (𝐴 − 𝐵 − 𝑏𝑁 + 𝑏𝑀 ) ≥ 0, ∆𝐸 + (10 + 𝐴 − 3𝐵 − 𝑏𝑁 + 𝑏𝐿 + 𝑏𝑀 ) ≥ 0,
4
2
2
4
2
2
1
1
1
1
1
1
𝑏𝐿 − 𝑏𝑀 ≥ 4𝐴 − 20, −∆𝐸 + (𝐴 − 𝐵 − 𝑏𝑀 + 𝑏𝑁 ) ≥ 0, −∆𝐸 + (10 + 𝐴 − 3𝐵 − 𝑏𝑀 + 𝑏𝐻 + 𝑏𝑁 ) ≥ 0,
4
2
2
4
2
2
1
1
1
1
1
1
𝑏𝐿 + 𝑏𝑀 ≤ 20, −∆𝐸 + (20 − 𝑏𝐿 − 𝑏𝑀 + 𝑏𝑁 ) ≥ 0, −∆𝐸 + (30 − 𝑏𝐿 − 𝑏𝑀 + 𝑏𝐻 + 𝑏𝑁 ) ≥ 0.
[
]
4
2
2
4
2
2
where 𝑏𝑘 is the equilibrium bid of type 𝑘 ∈ {𝐻, 𝑁, 𝑀, 𝐿} and ∆𝐸 = 𝐸𝑔𝑖𝑣𝑒 − 𝐸𝑡𝑎𝑘𝑒 . Precisely, in the
system of equations, given that a bidder with type H or N submits demand to take and a bidder with
type M or L submits demand to give (see the proof in section C.4.2.), the first and second lines are
participation constraints of type H, N, M and L respectively. For the third to six lines, each line is the
incentive compatible constraints of type H, N, M and L respectively.
[Table C-20 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue under imperfect information in the take-or-give auction with second-price payment, entry fees
1
and no sale condition with four discrete types (𝐸𝑔𝑖𝑣𝑒 = 𝐸𝑡𝑎𝑘𝑒 = 4 (15 + 𝐴))]
Type
realization
H
N
M
L
Equilibrium
strategy
Ex-ante outcome
(payment and allocation)
Participate with
10$ bid and
demand to take.
Participate with
0$ bid and
demand to take.
Participate with
0$ bid and
demand to give.
Each bidder pays the corresponding fee.
If both submit the same demand but
different bids, the highest-bid bidder
additionally pays equal to his opponent's
bid and the object is allocated according to
his demand.
If both submit the same demand and the
same bid, each bidder has 0.5 chance to
additionally pay equal to his opponent's bid
and the object is allocated according to his
demand.
If both submit different demands, there is
no additional payment and the demand
taker obtains the object.
Participate with
10$ bid and
demand to give.
Expected
bidder's
surplus
1
(15
4
Seller's
revenue
(from
both
bidders)
− 𝐴)$
0$
0$
1
(15
4
− 𝐴)$
1
(35 +
4
2𝐴)$
By solving the system, the type N and M are the binding types of the demand-to-take and demand-to1
give sides respectively; hence the seller designs 𝐸𝑔𝑖𝑣𝑒 = 𝐸𝑡𝑎𝑘𝑒 = 4 (15 + 𝐴). As a consequence, a bidder
with type N or M participates with submitting the corresponding fee and 0$ bid. For the equilibrium
strategies of the H- and L-type bidders, the similar arguments as presented in previous sections can be
applied. The equilibrium strategies, ex-ante outcome, expected bidder's surplus and seller's revenue are
summarized in Table C-20.
Notice that the auction extracts all surplus from the bidder with type N or M while it leaves the bidder
with type H or L some surplus as the information rents. The results show that the take-or-give auction
provides countervailing incentives. Precisely, by ordering the type from low (L) to high (H) we can see
that the rents are decreasing in type over one range (from type L to type M) and are increasing over
other range (from type N to type H).
C.5.2. Pooling equilibrium at middle-type range in the take-or-give auction with second-price
payment, entry fees and no sale condition
This section presents the seller's revenue when he designs the take-or-give auction with second-price
payment, entry fees and no sale condition with pooling equilibrium at middle-type range. According to
the case of four discrete types where the payoffs are specified in Table C-19, the seller intends to design
the auction which pools type N and M together (precisely, a bidder with type N or M will submit the
same doublet) and leaving type L and H be separated. Since this auction designs different rules than the
auction with separating equilibrium for the whole-type range, to distinguish from the previous auction
we call it "take-or-give auction with second-price payment, entry fees, no sale condition and middlepooling equilibrium."
Unlike the case of separating-equilibrium auction where the bidders are separated into two groups -demand-to-take and demand-to-give groups -- in the middle-pooling auction the seller wants to
separate bidders into three groups -- demand-to-take, demand-to-give and pooling groups -- by applying
his designed menu of entry fees. Specifically, the seller designs the menu (𝐸𝑔𝑖𝑣𝑒 , 𝐸𝑡𝑎𝑘𝑒 , 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 ) which
a bidder who demands to take will submit 𝐸𝑡𝑎𝑘𝑒 ; a bidder who demands to give will submit 𝐸𝑔𝑖𝑣𝑒 ; a
bidder who demands to be pooled will submit 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 .
Precisely, the pooling group means that any bidder who is in this group does not want to distinguish his
demand from other bidders who are in the group. Neither, the pooling bidder does not want to compete
for taking nor giving the object. In fact, which we will see later, a pooling bidder mostly participates to
prevent the no sale event. Hence, in other words, we may imagine the auction which since the no sale
condition all bidders participate to prevent the no sale event. However, some bidders who have high
willingness to pay will submit the doublet and compete for their desired allocations -- either to take or
to give; while other bidders who have low willingness to pay will be a kind of passive bidder who do not
compete for their desired allocations.
Hence, since there are two bidders in this setting, regardless of their bids, if both bidders submit
𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 the seller randomly allocates the object with 0.5 chance for each bidder; if there is one pooling
bidder and one taking bidder the taking bidder obtains the object; if there is one pooling bidder and one
giving bidder the giving bidder gives the object.
For the payment, a bidder pays the submitted fee and second price if the case is applicable. Precisely,
the second price is paid when two bidders submit the same fee. Also, if any bidder does not participate,
the auction is cancelled and both bidders pay nothing; the seller keeps the object.
[Table C-21 Illustrations of the payment and allocation mechanisms of the take-or-give auction with
second-price payment, entry fees, no sale condition and middle-pooling equilibrium]
Scenario
Strategies
Allocation
Payment
Both bidders submit 𝐸𝑡𝑎𝑘𝑒 .
Both pay 𝐸𝑡𝑎𝑘𝑒 .
10
Bidder i bids 10$.
Bidder i obtains it.
Bidder i additionally
Bidder j bids 7$.
pays 7$.
Both bidders submit 𝐸𝑔𝑖𝑣𝑒 .
Both pay 𝐸𝑔𝑖𝑣𝑒 .
Bidder i gives it.
11
Bidder i bids 10$.
Bidder i additionally
Bidder j is the obtainer.
Bidder j bids 7$.
pays 7$.
Bidder i submits 𝐸𝑡𝑎𝑘𝑒 and
10$ bid.
Bidder i pays 𝐸𝑡𝑎𝑘𝑒 .
12
Bidder i obtains it.
Bidder j submits 𝐸𝑔𝑖𝑣𝑒 and
Bidder j pays 𝐸𝑔𝑖𝑣𝑒 .
7$ bid.
Both bidders submit
Each bidder has 0.5 chance of being Both pay 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 and
𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 .
13
the obtainer.
no additional payment.
Both bid 0$.
Bidder i submits 𝐸𝑡𝑎𝑘𝑒 and
10$ bid.
Bidder i pays 𝐸𝑡𝑎𝑘𝑒 .
14
Bidder i obtains it.
Bidder j submits 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔
Bidder j pays 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 .
and 0$ bid.
Bidder i submits 𝐸𝑔𝑖𝑣𝑒 and
Bidder i pays 𝐸𝑔𝑖𝑣𝑒 .
10$ bid.
Bidder i gives it.
15
Bidder j is the obtainer.
Bidder j submits 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔
Bidder j pays 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 .
and 0$ bid.
Table C-21 illustrates the payment and allocation mechanisms in the auction with middle-pooling
equilibrium. The 10th-12th scenarios are the same as in the auction with separating equilibrium. The 13th15th scenarios present the case which some bidders submit 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 . Notice the 13th scenario where
both bidders submit 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 , since regardless of their bids the allocation is fixed by 0.5 chance of being
the obtainer for each bidder, there is no incentive for any bidder to compete with non-zero bid; hence
submitting zero bid is always optimal for a bidder who submit 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 . This also extends to the 14th and
15th scenarios.
Hence, the take-or-give auction with second-price payment, entry fees, no sale condition and middlepooling equilibrium is processed as follows:
1. Each bidder submits his doublet which composes of bid and entry fee -- 𝐸𝑡𝑎𝑘𝑒 if he demands
to take, 𝐸𝑔𝑖𝑣𝑒 if he demands to give or 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 if he demands to be pooled. Not
participating is also available which if any bidder does so both pay nothing and the
auction is cancelled.
2. If the auction is not cancelled, the seller selects the highest-bid bidder's demand and allocates
the object accordingly. If the bids are tied, the seller prefers to allocate it to a bidder
who pays 𝐸𝑡𝑎𝑘𝑒 than 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 and 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 than 𝐸𝑔𝑖𝑣𝑒 (and the transitivity holds).
3. The highest-bid bidder additionally pays equal to:
- his opponent 's bid if they submit the same entry fee,
- or zero if his opponent submits different entry fees or does not participate.
Under this setting, the seller designs the menu (𝐸𝑔𝑖𝑣𝑒 , 𝐸𝑡𝑎𝑘𝑒 , 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 ) which is characterized by the
following system of equations:
1
1
1
−𝐸𝑡𝑎𝑘𝑒 + (35 − 𝑏𝐻 − 2𝑏𝑁𝑀 ) ≥ 0, −𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 + (20 − 𝑏𝑁𝑀 ) ≥ 0,
4
2
4
1
1
1
−𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 + (20 − 𝑏𝑁𝑀 ) ≥ 0, −𝐸𝑔𝑖𝑣𝑒 + (35 − 𝑏𝐿 − 2𝑏𝑁𝑀 ) ≥ 0,
4
4
2
1
1
1
1
1
𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 − 𝐸𝑡𝑎𝑘𝑒 + (15 − 𝑏𝐻 − 𝑏𝑁𝑀 ) ≥ 0, 𝐸𝑔𝑖𝑣𝑒 − 𝐸𝑡𝑎𝑘𝑒 + (30 − 𝑏𝐻 + 𝑏𝐿 ) ≥ 0,
4
2
4
2
2
1
1
1
1
𝐸𝑡𝑎𝑘𝑒 − 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 + (15 − 3𝐴 + 𝑏𝐻 + 𝑏𝑁𝑀 ) ≥ 0, 𝐸𝑔𝑖𝑣𝑒 − 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 + (15 − 3𝐵 + 𝑏𝐿 + 𝑏𝑁𝑀 ) ≥ 0,
4
2
4
2
1
1
1
1
𝐸𝑡𝑎𝑘𝑒 − 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 + (15 − 3𝐵 + 𝑏𝐻 + 𝑏𝑁𝑀 ) ≥ 0, 𝐸𝑔𝑖𝑣𝑒 − 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 + (15 − 3𝐴 + 𝑏𝐿 + 𝑏𝑁𝑀 ) ≥ 0,
4
2
4
2
1
1
1
1
1
𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 − 𝐸𝑔𝑖𝑣𝑒 + (15 − 𝑏𝐿 − 𝑏𝑁𝑀 ) ≥ 0, 𝐸𝑡𝑎𝑘𝑒 − 𝐸𝑔𝑖𝑣𝑒 + (30 − 𝑏𝐿 + 𝑏𝐻 ) ≥ 0.
[
]
4
2
4
2
2
where 𝑏𝐻 is the equilibrium bid of a bidder with H type; 𝑏𝐿 is the equilibrium bid of a bidder with L
type; 𝑏𝑁𝑀 is the equilibrium bid of a bidder with N or M type (since the N and M type bidders are
pooled, at equilibrium they bid the same). Precisely, the first and second lines of the system present the
participation constraints of H, N, M and L types respectively. For the third to sixth lines, each line
presents the incentive compatible constraints of the type H, N M and L respectively.
Optimally, the seller designs 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 which is feasible for a bidder with N or M type (by satisfying the
participation constraints of the types) and designs 𝐸𝑡𝑎𝑘𝑒 and 𝐸𝑔𝑖𝑣𝑒 which are not feasible for a bidder
with N nor M type but are feasible for a bidder with H and L types respectively. Hence, the seller designs
𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 = 5 which is feasible and leaves no surplus for the bidder with N or M type; it also affects the
bidder to bid 𝑏𝑁𝑀 = 0.4 By solving the rests in the system, we get 𝐸𝑡𝑎𝑘𝑒 = 𝐸𝑔𝑖𝑣𝑒 =
35
;
4
and it affects
𝑏𝐻 = 𝑏𝐿 = 0. Table C-22 summarizes the results.
4
Even a bidder with some types in the pooling range has positive surplus, he submits 0$ bid since, according to the
auction rules, there is no incentive for him to compete with positive bid.
[Table C-22 Symmetric-equilibrium strategy, ex-ante outcome, expected bidder' surplus and seller's
revenue under imperfect information in the take-or-give auction with second-price payment, entry fees,
no sale condition and middle-pooling equilibrium (𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 = 5, 𝐸𝑔𝑖𝑣𝑒 = 𝐸𝑡𝑎𝑘𝑒 =
Type
realization
H
N
M
L
Equilibrium
strategy
Participate with
0$ bid and
𝐸𝑡𝑎𝑘𝑒 .
Participate with
0$ bid and
𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 .
Participate with
0$ bid and
𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 .
Participate with
0$ bid and
𝐸𝑔𝑖𝑣𝑒 .
Ex-ante outcome
(payment and allocation)
Expected
bidder's
surplus
35
)]
4
Seller's
revenue
(from
both
bidders)
0$
Each bidder pays the corresponding fee.
If both submit the same fee, each bidder
has 0.5 chance of being the obtainer.
If both submit different fees, the seller
allocates the object to a bidder who
submits 𝐸𝑡𝑎𝑘𝑒 than 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔 and 𝐸𝑝𝑜𝑜𝑙𝑖𝑛𝑔
than 𝐸𝑔𝑖𝑣𝑒 (the transitivity holds).
0$
55
$
4
0$
0$
Compare to the results in the auction with separating equilibrium (see Table C-20), the auction with
middle-pooling equilibrium less efficiently allocates the object but yields higher revenue (Proposition 1).
Intuitively, in the auction with middle-pooling equilibrium the seller reduces the information rents which
are paid to induce separating equilibrium (hence, it reduces the efficiency of allocation) and increases
his revenue.
PROPOSITION 1
In the take-or-give auction with second-price payment, entry fees and no sale condition, under some
circumstances, designing for middle-pooling equilibrium yields higher revenue than designing for
separating equilibrium for the whole-type range.
To see how the tradeoff between efficiency of allocation and revenue happens in this auction, notice
that to induce separating equilibrium the amount of information rent is increasing in the distance from
the binding type to a bidder's type; precisely, as presented in Table C-20, the N type is the binding type
1
on the demand-to-take extremity hence a bidder with H type gets information rent as 4 (15 − 𝐴)$
which is increasing in the distance from the binding type N -- which is precisely measured by 𝑊(𝐻) −
𝑊(𝑁) = 10 − 𝐴; while the M type is the binding type on the demand-to-give extremity, the rent paid to
a bidder with L type is increasing in the distance from the binding type M.
Hence, in the middle-pooling auction the seller pools some middle types together and selects the other
binding types on each extremity; precisely, as presented in Table C-22, (since there are only four types
and the two middle types are pooled) the H and L types are the binding types for demand-to-take and
demand-to-give extremities respectively. Then, the information rent paid to any type which is far
beyond the H and L types on each extremity is reduced when compared to selecting the N and M types
as binding types.
Precisely, suppose that we have one more type, say X is on the farthest of demand-to-take extremity.
The information rent paid to a bidder with X type to separate him from other types in middle-pooling
auction is less than in whole-type-separating auction.
C.6. Take-or-give auctions (continuous types)
As we have seen in the previous sections, any basic auction (like first- and second-price sealed-bid
auctions) fails to prevent the free-rider problem; hence we change the auction rules by providing the
proper incentives and find that the take-or-give auction with second-price payment fixes the problem
and is the efficient auction. Then, we increase the level of revenue by imposing additional rules -- entry
fees and no sale condition. We have seen that designing for the middle-pooling equilibrium in the
auction with the rules yields, under some circumstances, higher revenue than the whole-separatingequilibrium auction and seems to be the revenue-maximizing auction (see section C.5.).
In this section, we will formally explore the take-or-give auction under the continuous-type setting. It
composes as follows: first, the model is presented, section C.6.2. presents the take-or-give auction with
second-price payment which is the efficient auction, section C.6.3. presents the take-or-give auction
with second-price payment, entry fees, no sale condition and middle-pooling equilibrium which its
characterization is equivalent to the revenue-maximizing mechanism in Chen et al. (2010); hence it is
the revenue-maximizing auction.
C.6.1. Model
We focus on the same model as in Chen et al. (2010); there are two risk neutral and symmetric bidders
𝑁 = {1,2} who compete in an auction for an indivisible object with countervailing positive externalities
that has no value to the seller.
Denote 𝑖, 𝑗 ∈ 𝑁 and 𝑖 ≠ 𝑗 as the index of bidder. The 𝑖 𝑡ℎ bidder's type 𝑡𝑖 is randomly drawn from 𝑇 =
[𝑡, 𝑡] with distribution function 𝐹 and its associated density function 𝑓 which 𝑓(𝑡) > 0 for all 𝑡 ∈ 𝑇. As
presented in Fig. C-5 his payoff 𝑢𝑖 which depends on his type 𝑡𝑖 and the ex-post outcome is defined by
𝑊(𝑡𝑖 ) = 𝐴 + 𝑎𝑡𝑖 𝑖𝑓 ℎ𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑟
𝑢𝑖 (𝑡𝑖 ) = {𝐿(𝑡𝑖 ) = 𝐵 − 𝑏𝑡𝑖 𝑖𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑖𝑠 𝑡ℎ𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑟
0 𝑖𝑓 𝑡ℎ𝑒 𝑠𝑒𝑙𝑙𝑒𝑟 𝑘𝑒𝑒𝑝𝑠 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡
(1)
where 𝑎, 𝑏 > 0 and 𝐴, 𝐵 is high enough to make 𝑊(𝑡), 𝐿(𝑡) ≥ 0 for ∀𝑡 ∈ 𝑇. Assume that 𝐴 < 𝐵 which
yields a solution 𝑊(𝑡 ′ ) = 𝐿(𝑡 ′ ) (Assumption 3).
Assumption 3
𝐴 < 𝐵 and there is a solution 𝑊(𝑡 ′ ) = 𝐿(𝑡 ′ ).
utility
obtainer
non-obtainer
no sale
type
[Fig. C-5 An example of the bidder's utility function]
C.6.2. Take-or-give auction with second-price payment: the efficient auction
The take-or-give auction with second-price payment is processed as presented previously. Let each
bidder 𝑖 𝑡ℎ submit his doublet as (𝑑𝑖 , 𝑏𝑖 ) where 𝑏𝑖 ∈ ℝ is his bid and 𝑑𝑖 ∈ {𝑔𝑖𝑣𝑒, 𝑡𝑎𝑘𝑒} is his demand for
taking or giving the object.
Under the auction, the Proposition 2 presents the symmetric-equilibrium strategy. And the findings are
visualized in Fig. C-6 and C-7 for the strategy and ex-post allocation respectively. As concluded in the
Proposition 3, the auction efficiently allocates the object.
PROPOSITION 2
In the take-or-give auction with second-price payment, the symmetric-equilibrium strategy of the 𝑖 𝑡ℎ
bidder is
𝑔𝑖𝑣𝑒 𝑓𝑜𝑟 𝑡𝑖 ∈ [𝑡, 𝑡 ′ )
𝑑𝑖 (𝑡𝑖 ) = { 𝑡𝑎𝑘𝑒 𝑓𝑜𝑟 𝑡𝑖 ∈ (𝑡 ′ , 𝑡]
𝑑𝑖 ∈ [0,1] 𝑓𝑜𝑟 𝑡𝑖 = 𝑡 ′
(2)
𝑏𝑖 (𝑡𝑖 ) = |𝑊(𝑡𝑖 ) − 𝐿(𝑡𝑖 )|
(3)
bid
L2: bid=L-W
L1: bid=W-L
give t’ take
type
[Fig. C-6 Equilibrium strategy in the take-or-give auction with second-price payment]
[Fig. C-7 Allocation of the good in the take-or-give auction with second-price payment]
PROPOSITION 3
The take-or-give auction with second-price payment efficiently allocates the object.
C.6.3. Take-or-give auction with second-price payment, entry fees, no sale condition and
middle-pooling equilibrium: the revenue-maximizing auction
As presented in section C.5., this part characterizes the symmetric-equilibrium strategy under the takeor-give auction with second-price payment, entry fees, no sale condition and middle-pooling
equilibrium. Then, we compare the characterizations with of the optimal mechanism in Chen et al.
(2010); to see whether the proposed auction equivalently maximizes revenue.
Recall the process of the take-or-give auction with second-price payment, entry fees, no sale condition
and middle-pooling equilibrium, the seller designs the menu of entry fees (𝐹𝑡𝑎𝑘𝑒 , 𝐹𝑝𝑜𝑜𝑙 , 𝐹𝑔𝑖𝑣𝑒 ) which
some middle types optimally submit 𝐹𝑝𝑜𝑜𝑙 ; other types on the two extremities submit 𝐹𝑡𝑎𝑘𝑒 if they
demand to take and 𝐹𝑔𝑖𝑣𝑒 if they demand to give. If both bidder submits different fees the seller prefers
to allocate the object to a bidder who pays 𝐹𝑡𝑎𝑘𝑒 to 𝐹𝑝𝑜𝑜𝑙 and 𝐹𝑝𝑜𝑜𝑙𝑖𝑛𝑔 to 𝐹𝑔𝑖𝑣𝑒 (transitivity holds); each
bidder pays only the fee but does not pay the second price. If both bidder submits the same fee, the
object is allocated according to the highest-bid bidder's demand if the fees are 𝐹𝑡𝑎𝑘𝑒 or 𝐹𝑔𝑖𝑣𝑒 ; for the fee
𝐹𝑝𝑜𝑜𝑙 , the seller randomly allocates the object with equal probability for each bidder; each player pays
the fee and the highest-bid bidder pays the second price. If any bidder does not participate, the auction
is cancelled and both bidders pay nothing.
The auction which is characterized by the system of equations (𝐼𝐶 ∗ ), (𝐼𝑃∗ ) and (𝐵) (see in Appendix)
has the equilibrium as presented in the Proposition 4.
PROPOSITION 4
In the take-or-give auction with second-price payment, entry fees, no sale condition and middle-pooling
equilibrium, given that the auction is characterized by (𝐼𝐶 ∗ ), (𝐼𝑃∗ ) and (𝐵), in the equilibrium strategy
both bidders participate in the auction by submitting the doublet (𝐹𝑖∗ , 𝑏𝑖∗ ) which 𝐹𝑖∗ is entry fee and 𝑏𝑖∗ is
the bid as follows:
𝐹𝑔𝑖𝑣𝑒 𝑓𝑜𝑟 𝑡𝑖 ∈ [𝑡, 𝑡 ∗ ]
𝐹𝑖∗ (𝑡𝑖 ) = { 𝐹𝑡𝑎𝑘𝑒 𝑓𝑜𝑟 𝑡𝑖 ∈ [𝑡 ∗∗ , 𝑡]
𝐹𝑝𝑜𝑜𝑙𝑖𝑛𝑔 𝑓𝑜𝑟 𝑡𝑖 ∈ (𝑡 ∗ , 𝑡 ∗∗ )
𝑏𝑖∗ (𝑡𝑖 )
𝐿(𝑡𝑖 ) − 𝑊(𝑡𝑖 ) 𝑓𝑜𝑟 𝑡𝑖 ∈ [𝑡, 𝑡 ∗ ]
= {𝑊(𝑡𝑖 ) − 𝐿(𝑡𝑖 ) 𝑓𝑜𝑟 𝑡𝑖 ∈ [𝑡 ∗∗ , 𝑡]
0 𝑓𝑜𝑟 𝑡𝑖 ∈ (𝑡 ∗ , 𝑡 ∗∗ )
(4)
(5)
bid
L2: bid=L-W
L1: bid=W-L
L3: bid=0
t* tmin t**
Fgive Fpooling Ftake
type
[Fig. C-8 Equilibrium strategy in the take-or-give auction with second-price payment, entry fees, no sale
condition and middle-pooling equilibrium]
[Fig. C-9 Allocation of the object in the take-or-give auction with second-price payment, entry fees, no
sale condition and middle-pooling equilibrium]
Moreover, Fig. C-8 and C-9 visualize the equilibrium strategy and allocation respectively. The auction
gets similar results as of the optimal mechanism by Chen et al. (2010). In the Appendix, by comparing
the characterizations, the proposed auction equivalently maximizes revenue (in Proposition 5).
PROPOSITION 5
The take-or-give auction with second-price payment, entry fees, no sale condition and middle-pooling
equilibrium maximizes revenue.
Here, it is worth to notice some complications of the auction. Since the auction has three entry fees, its
applicability may not good. Moreover, under some circumstances, theoretically, negative bids are
possible which in practice it means the seller subsidizes some bidders; however, it seems to be less
applicable as well.
Even the revenue-maximizing auction seems to be less applicable, the result shades the light that any
auction which induces separating equilibrium for the whole-type range should not maximize revenue
since the seller pays too much rent. An auction which induces pooling-separating-mixed equilibrium
should be the family of revenue-maximizing auction.
C.7. Concluding remarks
This study introduces a new auction called "take-or-give auction." Since the new auction provides
proper incentives to both high-type bidders who want to obtain and consume the object and low-type
bidders who want others obtain and consume it, the auction is the best auction -- either for efficient
allocation or revenue maximizing -- to be implemented in the case of auction for an object with
countervailing positive externalities.
For one who concerns the efficient allocation (e.g., government), as presented in section C.6.2., the
take-or-give auction with second-price payment is optimal. While if one concerns the revenue
maximization (e.g. any profit-maximizing agent), the take-or-give auction with second-price payment,
entry fees, no sale condition and middle-pooling equilibrium is optimal.
For further study, one may design other optimal auction which is more practically implemented (as
mentioned in section C.6.3. that the auction with middle-pooling equilibrium in this study has some
complications which lead to less applicability). Or one may extend the model to, for instance, 𝑛-bidder
case or introducing the coalition-proof constraint into the seller's problem.
C.8. References
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International Economics, 73, 309-332.
Brocas, I. (2007). Auctions with Type-Dependent and Negative Externalities: The Optimal Mechanism.
Mimeo, USC.
Chen, B., & Potipiti, T. (2010, September). Optimal selling mechanisms with countervailing positive
externalities and an application to tradable retaliation in the WTO. Journal of Mathematical
Economics, 46(5), 825-843.
Jehiel, P., & Moldovanu, B. (2000). Auctions with downstream interaction among buyers. The RAND
Journal of Economics, 31(4), 768-791.
Jehiel, P., Moldovanu, B., & Stacchetti, E. (1996). How (not) to sell nuclear weapons. The American
Economic Review, 86(4), 814-829.
Lewis, T. R., & Sappington, D. E. (1989). Countervailing Incentives in Agency Problems. Journal of
Economic Theory, 294-313.
Myerson, R. B. (1981). Optimal Auction Design. Mathematics of Operations Research, 58-73.
C.9. Appendix
(Provided upon request)