We propose a model of sequential bidding for a valuable object, such as a takeover target, when it is costly submit or revise a bid. An implication of the model is that bidding occurs in repeated jumps, a pattern that is consistent with certain types of natural auctions such as takeover contests. The jumps in bid communicate bidders' information rapidly, leading to contests that are completed with a small number of bids. The model provides several new results concerning revenue and efficiency relationships between different auctions, and provides an information-based interpretation of delays in bidding. Prepublication version available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=161013

1. Electronic copy available at: https://ssrn.com/abstract=3181599
June 29, 1998 Comments Welcome
seqsld.tex
A Theory of Costly Sequential Bidding
Kent Daniel
Graduate School of Business
University of Chicago
David Hirshleifer
School of Business Administration
University of Michigan

2. Electronic copy available at: https://ssrn.com/abstract=3181599
* Why Bidder-Profit-Equivalence Result is Surprising
We did not rig up assumptions in order to concoct a neat result. We picked what we
thought were ‘natural’ (though stylized) assumptions, and thought we had a complicated
and non-comparable auction.
• stochastically realized deadweight bid costs
• bid schedules ‘top out’, so can’t use envelope theorem in these ranges
• each bidder has an infinite family of foreseen bid schedules as FTB after any
number of delays, and also his bid schedule as STB. Which bid schedule the bidder
applies depends stochastically on the behavior of the other bidder.
* Heuristic Intuition for Bidder-Profit-Equivalence Result
1. When interior solution FTB schedule, usual envelope condition implies F(θ).
2. When corner solution FTB schedule, F and b really are constants (instead of
differentiating as if they were constants), get same derivative F(θ).
3. Whenever you are STB, your profits are just like in ratchet solution, i.e., in a
second price auction (with lower bound on other guy’s valuation of γ). Thus, it is
like being a bidder in an auction where can use envelope condition to get derivative
F(θ)
Problem: the various schedules etc. are based on various different conditional distri-
butions different spots of the game tree.
1

3. Overview
• Preemptive Bidding
in Sequential Auctions
• The Basic Setting
• The Signalling Equilibrium: Zero Bid Costs
– Equilibrium Description and Derivation
– The Signalling Schedule
– Comparison with Sealed-Bid First-Price Auction
• A Signalling Equilibrium: Positive Bid Costs
– Bidding Schedule after Delays
– Payoff to Defections
– Comparison with First Price Sealed Bid Auctions with
Costs or a Minimum Bid
• An Alternative “Credulous” Equilibrium
– More Credulous FB Beliefs
– Isomorphic Maximization Problem
• Extensions
2

4. – New Information Arrival
– Multiple Bidders, Stochastic Arrival of Bidders,
Correlated Valuations
• Conclusions
3

5. Sequential Bidding
• Do bidders learn through bidding process?
• Do high bids intimidate competitors?
• In practice, why does bidding occur in big jumps?
• Does it ever pay to delay and observe competitors?
• Are sequential auctions with jumps optimal for the seller?
• How do sequential auctions relate to static auctions?
4

6. Preemptive Bidding in Sequential Auctions
Standard theory of sequential bidding:
The Ratchet Solution.
• Costless bidding.
• Bids always increase by the minimum allowed increment.
• Auction ends after many rounds of bidding.
• Price = second-highest valuation.
• Bidders never delay to learn from the bidding process.
In practice:
(“Natural” Auctions: takeovers, houses, used cars)
• Often large increments in each round.
• Often only a few rounds of bidding.
• Bidders often delay.
Possible explanation:
• Sequential Signalling.
• Each bidder jumps the bid substantially to intimidate oppo-
nents into quitting early.
5

7. Our Paper:
• Sequential Bidding as a Learning Process
• Ratchet Solution minimizes rate of learning
• Fully revealing bids maximize rate of learning.
• Our equilibria: maximize rate of relevant learning
– First bid is fully revealing
– SB either quits or reveals just enough to prove he will
win.
Why not just ratchet up?
• Symmetric Information Example.
– θ1 = 100, θ2 = 80.
– Zero bid costs:
∗ Object sold from 0 to 80.
∗ Weak dominance or trembles:
Object sold for 80.
– Positive Bid Costs: (Hirshleifer/Png 1990)
∗ Object sold for 0.
• Asymmetric Information.
– Signalling ⇒ loser should quit.
– Strong equilibria if bidding is costly.
6

8. Is bidding costly?
• Takeovers:
– Opportunity cost of time and attention.
– Costs of legal counsel.
– Investment banking fees.
– SEC filings and administrative costs.
– Coordinating operations with the transaction.
– Opportunity cost of extending funds commitment.
• Other Auctions:
– Opportunity cost of time and attention.
– Opportunity cost of extending funds commitment.
7

9. Related Papers on Takeovers and
Sequential Bidding
• Fishman (1988, Rand Journal)
– Two bidders.
– Private valuations.
– SB must pay investigation cost to learn valuation.
– If SB investigates, ratchet solution.
– FB bids 0 or high preemptive bid.
• Hirshleifer/Png (1990, Review of Financial Studies)
– Investigation costs and bidding costs in discrete example.
– Quit when you’re beat.
• Bhattacharyya (1990) (2 Models)
– Investigation Cost
– Uninformative entry fee, followed by ratchet solution.
– FB bids to signal his valuation.
• Also on jump bidding: Avery (1992)
– Correlated Valuations
– Binary choice for initial bid, sometimes bid high.
– First bid is used to coordinate among multiple equilibria.
8

10. Implications
• Explains bidding wars with sequences of jumps in bid.
• Delays may occur before bidding begins.
– A failure to bid is informative
– A weak bidder waits to assess strength of competition.
– An arbitrarily large amount of delay can occur.
• Even if investigation is costless, object may not be sold to the
highest valuation bidder.
– Even a bidder with valuation θ = θ − ǫ, after reveal-
ing his valuation, may be “psyched out” by lower-valued
opponent.
• The sequential bidding auction is asymptotically optimal as
bid cost approaches zero.
• Bid schedule equivalence to FPSB auction with entry fee.
• Bidder expected profit equivalence to FPSB auction with
minimum bid.
• Seller’s expected profit is lower than in FPSB auction by ex-
pected bid costs incurred.
9

11. The Basic Setting
Assumptions:
• n risk neutral bidders with independently distributed private
valuations θi.
(n = 2)
• Can’t credibly reveal valuation except through the bidding.
• Distribution of valuations Fi(θ) on [θ, θ]. (symmetric here)
• Order of moves predetermined.
• Bidders may pass or bid.
• Bidding cost of γ is paid whenever individual bids,
not when he passes.
• Auction ends after a bid is followed by a sequence of n − 1
passes. Final bidder wins and pays his final bid.
• Bidders maximize expected profit
net of bidding costs incurred.
10

12. Equilibrium Description
γ = 0
1. FB makes a bid b1(θ1) that signals his valuation.
2. SB’s equilibrium strategy:
Bid ˆθ1 if θ2 > ˆθ1
Pass if θ2 ≤ ˆθ1.
3. ⇒ highest valuation wins.
4. SB, if he wins, pays θ1, same as in ratchet solution.
5. Auction always ends after SB’s move.
6. FB bids b1 = E [θ2|θ2 < θ1].
7. FB pays average price he’d pay in Second Price auction (or
Ratchet Solution).
8. SB pays same price he’d pay in Second Price auction (or
Ratchet Solution).
9. ⇒ bidder expected profits, seller expected revenues the same
as ratchet solution.
10. Order of moves indifference: π1(θ) = π2(θ) for all θ.
11

13. Derivation of the Equilibrium
γ = 0
• Equilibrium Conjectures:
1. Fully revealing FB schedule b(θ).
2. FB plans to bid only once
(if SB adheres to equilibrium behavior).
• By 2., FB solves:
π(θ) = max
b
(θ − b)F(ˆθ(b)).
• Revealing bid ⇒ equivalent choice of signalled valuation:
max
ˆθ
F(ˆθ)[θ − b(ˆθ)].
– Identical to FPSB auction problem
– ⇒ FB bid schedule identical.
• Differentiate w.r.t ˆθ, equate to zero, set ˆθ = θ. FOLDE:
b′
(θ) + b(θ)
f(θ)
F(θ)
− θ
f(θ)
F(θ)
= 0.
• General solution (for θ > θ):
b(θ) =
1
F(θ)
θ
θ∗ sf(s)ds + bF(θ∗
) .
• Unique solution (b(θ) = θ) is:
b(θ1) =
θ1
θ tf(t)dt
F(θ1)
= E[θ2|θ2 < θ1].
12

14. Envelope Condition and FB Profits
• Auction “revenue” equivalence technique (see Milgrom/Weber
1982).
• FB’s expected profit function:
π(θ) = max
b
(θ − b)F(ˆθ(b)).
• Envelope theorem:
dπ
dθ
= F(θ).
because b is optimized and since ˆθ(b) = θ in equilibrium.
• Integrating:
π(θ) =
θ
θ
F(s)ds. (1)
– (imposing initial condition π(θ) = 0).
13

15. Uniform Distribution Example
• If θ1, θ2 ∼ U[0, 1]:
f(s) = 1, F(s) = s.
b(θ) = E[s|s < θ] =
θ
2
.
ˆθ(b) = 2b
• Geometric illustration.
14

16. Profitability of Defections
• What about conjecture FB plans to bid just once?
• If FB bids above b(θ), won’t bid again
– ⇒ no overbids.
• Alternative:
1. Bid low.
2. If SB bids, counter with higher bid.
• Intuition: If defect low, ˆθ < θ:
– gain when θ2 < ˆθ.
– lose when ˆθ < θ2 < θ.
• On average these precisely offset!
15

17. • Suppose FB’s second bid signals his true valuation.
• FB’s second-bid schedule:
– same as for the first, except:
– FB knows that θSB ∈ (ˆθ, θ] ⇒
– FB’s posterior beliefs about SB are
F†(s) = Pr(θ2 < s|θ2 > ˆθ) (2)
– Boundary condition:
∗ Minimum bid ˆθ.
∗ Made by FB with θ = ˆθ.
– So counterbid schedule is:
2b1(θ1) = E θ2|ˆθ1 < θ2 < θ1 .
• FB’s expected profit from the defection strategy:
π2(ˆθ, θ) = F(ˆθ) θ − b(ˆθ) + [F(θ) − F(ˆθ)] θ − 2b1(θ, ˆθ) .
• Substituting for the bid schedules, this reduces to same profit
as equilibrium strategy.
– Same as equilibrium strategy.
• Second bid problem isomorphic first
⇒ expected profit from any finite defection ending with truth-
ful signal is the same.
• Never truthfully signal ⇒ strictly worse.
16

18. Costly Bidding: γ > 0
• Each bidder pays γ each time he bids, zero if he passes.
• FB, SB versus FTB, STB
• Lower cutoff θ∗
for FB to bid at move 1.
• His initial bid schedule: (θ > θ∗
)
b(θ) =
θ
θ∗ tf(t)dt
F(θ)
.
where θ∗
is defined by:
θ∗
F(θ∗
) − γ =
θ∗
γ
F(s)ds.
– LHS = expected profit from lowest type bidding.
– RHS = expected profit from passing.
• Same FOLDE as before. Boundary condition:
– Lower limit of integral comes from lowest possible bidder
θ∗
.
– This type bids zero (see below).
• Same FODE as in FPSB auction.
• FB’s bid schedule is identical to bidders in FPSB auction;
entry fee matches valuation of lowest bidder.
– Entry fee > γ.
– Cutoff higher because waiting is profitable.
17

19. • If FB bids:
– SB bids ˆθ1 − γ if θ2 ≥ θ1.
– Otherwise passes.
• Skeptical FB beliefs:
– If SB defects to bid b2 < ˆθ1 − γ, FB thinks SB has
valuation b2 + γ.
– SB defection unprofitable.
• FB bid of zero signals θ∗
> θ, so sometimes wins, makes
money.
18

20. Equilibrium Bidding Sequence
Bid Cutoff Sequence Diagram
• Suppose FB passes.
• SB knows θ1 < θ∗
1.
• So θ∗
2 < θ∗
1.
– Similarly, θ∗
n+1 < θ∗
n,
– limn→∞ θ∗
n = γ.
• After FB passes, SB bids according to a similar schedule
except:
– Use distribution conditional on θ1 < θ∗
1.
– Top-Out: A signal θ∗
1 always wins.
• SB’s equilibrium bid schedule as FTB follows same differen-
tial equation:
– Different distribution on opponent
– Different boundary condition.
• FB response:
– FB bids ˆθ2 − γ if θ1 ≥ θF .
– Otherwise pass.
• Highest-valuation bidder wins.
19

21. • See Figure 3: The First Four Bidding Schedules...
• See Diagram: “Equilibrium Bid Sequence: γ > 0”
• High valuation bidder always wins.
• FTB’s Equilibrium Bid Schedule, n’th move
bn(θF ) =



θF
θ∗
n
sf(s)
F(θF ) ds if θF ≤ θ∗
n−1
θ∗
n−1
θ∗
n
sf(s)ds
F(θ∗
n−1) if θF > θ∗
n−1,
(3)
where F(·), f(·) are prior distribution, density functions for
STB’s valuation.
20

22. Analysis of Defections
• FTB does strictly worse if he:
– Defects to a different bid level.
– Defects to a different amount of delay.
• Envelope Condition Method
• Delay or underbidding “drives out” lower FTB types.
21

23. Determination of Payoff to Defections
- Low Bid Strategy -
• Overbid defection obviously unprofitable.
• Consider the following (out-of-equilibrium) strategy for FTB:
– Initially bid to signal valuation as ˆθ < θF .
– If STB responds (with bS = ˆθ −γ), FTB rebids to truth-
fully reveal his valuation as θF .
– Bid based on conditional STB valuation distribution over
[ˆθ, θ]
• STB’s response to FTB’s second bid is to
Bid θF − γ if θS >2
ˆθF
Pass if θS <2
ˆθF
– This response is “generous” to the possible defection.
• Payoff to two-bid strategy < equilibrium payoff:
π(ˆθ, θ) = F(ˆθ) θ − bF (ˆθ) − γ
E(first bid gain)
+
F(θ) − F(ˆθ) θ − 2bF (θ, ˆθ) − 1 − F(ˆθ) γ
E(second bid gain)
• Alternative:
• Let θ∗
F be the equilibrium lowest valuation type willing to
become FTB at current move.
22

24. • Let G(·) be conditional distribution for STB.
• In equilibrium, integrating the profit derivative G(θ), get
π(θ) = π(θ∗
F ) +
θ
θ∗
F
G(s)ds.
• Expected profit from defecting to signal ˆθ < θ,
πD
(θ; ˆθ) = π(θ∗
F ) +
ˆθ
θ∗
F
G(s)ds + (θ − ˆθ)G ˆθ (4)
+Pr(θS > ˆθ)
θ
θ′ G(s|θS > ˆθ)ds.
• Here θ′
is the lowest θ such that FTB is just willing to counter
STB’s bid of ˆθ − γ.
• θ′
> ˆθ, because a low FTB’s counter would probably lose.
• Last term: expected profit from rebidding
• Underbidding introduces a “lapse” in the G(·) integral.
• ⇒ strictly lower profits.
• Second bid problem isomorphic to first bid problem. Iterative
argument: any finite defection is sub-optimal.
– Infinite pattern of defection is sub-optimal.
23

25. Defection by Passing
• A bidder can pass planning to bid again.
• We consider a defection where the player:
– Initially passes.
– Bids so as to signal his correct valuation in the next round.
• The expected payoff from this defection is strictly less than
from the equilibrium strategy.
• Equilibrium expected profit for FB (say) is
πB(θ) = πB(θ∗
1) +
θ
θ∗
1
F(s)ds. (5)
• Suppose θ close to θ∗
1.
– After SB bids, FB will not counter.
– Defection unprofitable.
• There exists a θ∗
P > θ∗
1 just willing to counter SB.
• Suppose θ > θ∗
P .
• Not hard to show that profits are:
πP (θ) = πP (θ∗
1) + (θ − θ∗
1) · F(θ∗
1) +
θ
θ∗
P
F(s)ds.
• Losing money on the gap (θ∗
1, θ∗
P ) where FB would not want
to counter.
24

26. Bidder Profit Equivalence
First and second bidder profits in CSB auction identical to
FPSB auction with minimum bid γ, i.e.
π(θ) =
θ
γ
F(s)ds.
• At any stage, FTB’s expected profit function:
π(θ) = max
b
(θ − b)G(ˆθ(b))
where G is conditional distribution function of STB.
• Envelope theorem: In revealing region of bid schedule:
dπ
dθ
= G(θ).
because b is optimized and since ˆθ(b) = θ in equilibrium.
• When θ is in flat part of bid schedule,
dπ
dθ
= 1,
still works.
• For STB, like second price auction (pay opponent’s valua-
tion).
25

27. • Since:
– Object allocated efficiently;
– Bidder profits are identical
– Bid costs incurred;
• Seller does worse in CSB auction by expected bid costs.
• Losses are less than 2γ.
• As γ → 0, auction is efficient.
26

28. The Credulous Equilibrium
FB:
Can beat θ2 = ¯θ − ǫ.
Bluffing, inefficient outcome.
SB:
Two-sided bluffing.
As γ → 0:
Still approaches efficiency, and maximal seller revenue.
27

29. Determination of Cutoff Level Sequence
• θ∗
i < θ∗
i−1, and that limi→∞ θ∗
i = γ
• To derive the sequence of cutoff levels, find the bidder who is
indifferent between bidding zero now and passing.
• Assume that after n − 1 passes, FB with valuation θ∗
n is
considering whether to bid or not.
– If he bids zero now, his expected profit is
πB = θ∗
n·Pr(θ∗
n > θS|θS < θ∗
n−1)−γ = θ∗
n



F(θ∗
n)
F(θ∗
n−1)


−γ.
– If he passes this round, by bidder-profit equivalence, his
profit is
θ∗
n
γ
F(s|s < θ∗
n−1)ds.
Equating determines θ∗
n.
• – Graph illustrating cutoff levels –
28

30. The Credulous Equilibrium
- Isomorphic Maximization Problem -
• SB responds to FB’s signalled valuation θ1 with a bid of
b2D(θ1) if his valuation is greater than θ2D ≡ b2D + γ.
• θ2D satisfies the following two conditions:
1. FB’s optimal second bid yields zero expected profits, given
FB’s credulous beliefs:



F(2b1 + γ) − F(θ2D)
1 − F(θ2D)


 (θ1 − 2b1) − γ = 0.
2. 2b1 is chosen optimally (obtained by differentiating this
w.r.t 2b1):
f(2b1 + γ)(θ1 − 2b1) − F(2b1 + γ) − F(θ2D) = 0
– This means that we can write θ2D as a function of θ1
• Given this, FB’s maximization problem is:
max
ˆθ1
π(ˆθ1) = F(θ2D(ˆθ1))[θ1 − b(ˆθ1)] − γ.
• Define H(ˆθ) ≡ F(θ2D(ˆθ)) – the probability that SB will be
deterred by a signalled valuation of θ.
29

31. • As before, gives LFODE with solution:
b(θ1) =
θ1
θ∗ t · h(t)dt
H(θ1)
,
where
γ(H(θ∗
) − H(γ)) =
θ∗
γ
t · h(t)dt.
30

32. Credulous Equilibrium Description
1. FB:
Passes if θ1 ≤ θ∗
Bids b1(θ1) =
θ1
θ∗ t · h(t)dt
H(θ1)
if θ∗
< θ1 ≤ ˆθ
where
γ(H(θ∗
) − H(γ)) =
θ∗
γ
t · h(t)dt.
and
H(θ) ≡ F(θ2D(θ))
, Probability that SB will be deterred with a bid of θ.
2. SB:
Passes if θ2 < θ2D(ˆθ1)
Bids b2D = θ2D − γ otherwise
• b2D is endogenously determined to be such that, under
credulous beliefs, FB’s expected profit from rebidding is
zero.
• FB believes that if he rebids only SB types θ2 ∈ [θ2D, 2b1+
γ] will drop out of bidding
• Note that FB, if he bids, will not always win the auction,
even if he has the highest valuation
– Standard revenue equivalence results do not apply.
31

33. Extensions
• Multiple Bidders
• Stochastic Arrival of Bidders
• Correlated Valuations
• Modified Information Arrival Process
32

34. Summary
• Sequential bidding as a learning process
• Sequential signaling, bluffing
• Why bidding proceeds in a few substantial jumps
• Why bidders wait
• New auction optimality and equivalence results
33