Operations Research Letters 42 (2014) 12–15

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Operations Research Letters

journal homepage: www.elsevier.com/locate/orl

An auction with positive externality and possible application toovertime rules in football, soccer, and chessDaniel Granot a,∗, Yigal Gerchak b

a Sauder School of Business, University of British Columbia, Vancouver, B.C., Canadab Department of Industrial Engineering, Tel Aviv University, Tel Aviv, Israel

a r t i c l e i n f o

Article history:Received 12 June 2013Received in revised form1 November 2013Accepted 1 November 2013Available online 9 November 2013

Keywords:First-price auctionSecond-price auctionTruthful biddingFootballSoccerChess

a b s t r a c t

We analyze auctions with positive externality, wherein the utility of each player who submitted a losingbid is strictly increasing in the price paid by the winning bidder. Such an auction was recently proposedfor determining the starting team and the starting yard line in an overtime period in American football.We analyze the NFL case and also consider other football leagues, as well as tie-breaking by penalty shotsin soccer, and overcoming a draw situation in chess.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The fast-growing role of auctions in the economy is primarilydue to their effective allocative properties (see, e.g., [8,9]). Sometypes of auctions inherently encourage ‘‘true bids’’, reflecting thebidders’ true valuations, which are economically efficient. Indeed,the well-known economic advantage of a second-price auctionover a first-price one is that the former induces truthful bids, whilethe latter does not [12]. Inmost common types of auctions, the bid-ders who do not win are not influenced by the magnitude of thewinning bid. One exception is a ‘‘knockout auction’’ (see, e.g., [4,3]),used for the allocation of jointly owned indivisible items, wherethe winner pays the losers their shares of the winning bid (there isno seller). Amotivation for the presentwork is another type of auc-tion, recently proposed in [1] for determining the starting teamandstarting position in the overtime period inNational Football League(NFL) games. Instead of determining the starting team by a cointoss, as is currently practiced, in the proposed auction the teamscompete over who will be the starting team by submitting bidsfor the starting position in a manner to be explained in the sequel.We also consider bidding possibilities for starting overtime peri-ods in NCAA and Canadian football, tie-breaking by penalty kicks in

∗ Corresponding author.E-mail addresses: daniel.granot@sauder.ubc.ca (D. Granot), yigal@post.tau.ac.il

(Y. Gerchak).

0167-6377/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.orl.2013.11.001

soccer, for whichwe provide amore formal analysis, and overcom-ing a draw outcome in chess competitions. We show, for example,that the auctionmechanism proposed in [1] to determine the teamthat will be first on offense at the start of the overtime period inthe NFL will not induce truthful bidding, and that first-price andsecond-price auction mechanisms in these situations will induceunderbidding and overbidding, respectively, by the teams.

2. Bidding with externalities

In most types of auctions, the losing bidders do not incur anypost-bid externalities. However, a negative externality, say, couldoccur to a losing bidder if a win in the auction by a rival playerresults in a decrease in hismarket power ormarket share. Consideran auction for an item with two bidders, and assume that the truevaluation for the item for bidder i is vi, i = 1, 2, but a loss by bidderi has a negative externality to him of value ui, i = 1, 2, where ui ≥

0. Then, it can be easily argued that the true valuation of bidder iis vi + ui, i = 1, 2, and, for example, in a second-price sealed-bidsauction, the players will bid their true valuations. This observationcan be generalized to many bidders. Suppose the value of anauctioned item for bidder i is vi, his externality cost if player j, j = i,wins the item is uj, uj ≥ 0, and assume the existence of a vector, P i,providing the prior conditional probability that player j, for all j = i,will win the item, given that player i did not win the auction. Then,for example, if bidder i is risk neutral, the expected externality costto bidder i if he does not win the item is Ei ≡ Σk=iP i

kuk, and in asecond-price auction, bidder i will bid his true valuation, vi + Ei.

D. Granot, Y. Gerchak / Operations Research Letters 42 (2014) 12–15 13

Next, consider the following scenario, where two players, 1 and2, submit sealed bids, bi, i = 1, 2, respectively, and the playerwith the highest bid (ignoring ties) wins the item and pays for it aweighted average, αb1 + (1−α)b2, of the two bids, for 0 ≤ α ≤ 1.Each of the players would like to own the item but we will assumethat in this scenario, if a player loses, because he submitted thelower bid, he strictly prefers that the winning player pays morefor the item rather than less. That is, we will consider in this noteauction situations, which we will refer to as auctions with positiveexternality, in which the utility function of the losing bidder strictlyincreases with the price for the item paid by the winning bidder.

Note that knockout auctions can be naturally viewed as auc-tions with positive externality of the type discussed above. Indeed,losing bidders in a knockout auction strictly prefer the winning bidto be as large as possible, since a higher winning bid increases thepayment they receive from their share of the indivisible object.

3. Application to overtime rules in football, chess, and soccer

Overtime in the National Football league (NFL)In the NFL, a tied football game goes into a sudden death over-

time period. In regular season games, the overtime period is 15minand the game can end up in a tie. In playoff games, there must be awinner, and the game continues until someone scores. If the start-ing team in the overtime period scores a touchdown on its firstpossession, the game is over, and the scoring team is the winner.If the starting team only scores a field goal or fails to score on itsfirst possession, the other team gets possession of the ball (after akickoff if a field goal was scored), and has a chance to tie the game,if the first team scored a field goal, orwin the game. If the first teamscored a field goal on its first possession and the second team failedto tie the game on its first possession, the starting team is the win-ner. Otherwise, the game continues until one of the team scores(a touchdown or a field goal) or, in regular season games, until theperiod is over, inwhich case the gameendswith a tie. The key ques-tion is how does the overtime period start—which team is first onoffense and from where (distance to the end zone) it starts. Natu-rally, the team which is first on offense has a clear advantage. If itscores a touchdown, the game is over. The current rule is to deter-mine possession by a coin toss, and have a kickoff. However, manyin football seem to be uncomfortable to base such an important de-cision on a coin toss ([1] and media quotes therein). While a cointoss is fair ex-ante, it is not ‘‘fair’’ ex-post. The loser in the coin tosswould like to be thewinner. Indeed, according to [2], of the 24 non-tied overtime games played so far under the new overtime rulesintroduced by the NFL (field-goal exclusion rule), 16 (2/3) werewon by the receiving team and eight (1/3) by the kicking team.In other words, the winner of the coin toss enjoyed, on average, a2:1 edge in overtime. To overcome the difficulties stemming froma coin toss, an auction mechanism is suggested in [1] according towhich each team submits a bid for the yard line from which playstarts. The yard line selected will be the average of the two bids,with the lowbidder playing defense and the high bidder playing of-fense. To illustrate, suppose Team A1 submits a bid of 75 yard line,meaning it is prepared to start from its own 25 yard line if it is thefirst team to get the ball, and Team A2 submits a bid of 70 yard line,then Team A1 will start on offense, Team A2 will start on defense,and the game will start 72.5 yards away from Team A2’s end zone.

A true valuation, vi, of Team Ai in the above football example isthe furthest distance from Team Aj’s, j = i, end zone it is preparedto start from, if it is the first team on offense. At exactly vi yardsaway from its opponent’s end zone, it is indifferent whether it getsto be the first teamon offense or be on defense, 100−vi yards awayfrom its own end zone. However, note that if a team loses the bid, itstrictly prefers that the other team will start further away from itsown end zone, which casts the football bidding case as an auctionwith externality discussed above.

Overtime in American college football and Canadian footballAs in the NFL, in American college football, if the teams are tied

at the conclusion of regulation time, an overtime period ensues.The overtime period begins with a coin toss to determine whogets possession and who defends their goal first. Teams who winthe coin toss overwhelmingly prefer to defend their goal first (see,e.g., [11,1]). Unlike overtime in the NFL, college football overtimeallows each team the chance to have possession. The team that getspossession first receives the ball on its opponent’s 25-yard line.The team can keep possession of the ball until it either: (i) scores atouchdown, (ii) attempts a field goal, (iii) runs out of downs, or (iv)turns over the ball. Once the first team’s possession is over, the sec-ond team gets possession and follows the same format. If the firstteam scored a touchdown and an extra point, then the second teammust do so in order for the overtime period to continue. If not, thegame is over when the second team loses possession. If both teamsscore the same amount of points, a second round of overtime isplayed following another coin toss to determine who goes first. Ifa third overtime is needed, then the teams are forced to attempta 2-point conversion following a touchdown. They cannot kick anextra point.

Since teams in American college football overwhelminglyprefer to go second [11,1], it is ex-post fairer to let teams bid whogoes second, rather than let a coin toss determine this outcome.Teams could then submit bids as to the distance, beyond the 25yard line, they are prepared to start from, if they go second. Forexample, if Team A and Team B submit bids bA and bB, respectively,and bA > bB, then Team A goes second, starting, in a first-priceauction, from the 25 + bA yard line, while Team B goes first, andstarts from the 25 yard line. As in the NFL, the team that submittedthe losing bid strictly prefers that the team submitting thewinningbidwill start further away from its goal line, which implies that theauction as to who goes second in overtime in college football is anauction with externality discussed above.

Overtime rules in the Canadian Football League (CFL) are similarto those in American college football (with one of the differencesbeing that teams start from the 35 yard line rather than the 25yard line). Teams prefer to go second rather than first in overtimeand a resolution who goes first can be determined by bidding in anauction with externality of the type described above.Overtime in chess

In the 2010US Chess Championship Competition, the final stageof the regular competition between the last two contestants wasa draw, and the players moved to an ‘‘Armageddon’’ Stage. In theArmageddon Stage, the player using the black pieces only has toend the game in a draw to win, while the player using the whitepieces has to earn an outright victory. If one of the players needsa draw to win, they can play more conservatively. It makes thema heavy favorite even with black pieces. To decide who gets whichside, the players bid time off their sixty-minute game clock. In the2010USChess Championship Competition, the two remaining con-testants were Yury Shulman and Gata Kamsky. Without knowinghow much time their opponent was willing to sacrifice, Shulmanbid 21 min off the clock and Kamsky offered 35 min. A first-priceauction mechanism was implemented by the organizers, and ac-cordingly, it left Kamsky with just 25 min to complete every movein the game as black versus the sixty minutes that Schulman hadto complete his moves with the white pieces. The game ended in adraw, and Kamsky won (see, e.g., [5]).

Clearly, bidding to play black for a draw in the ArmageddonStage in chess is an auction with externality of the type consideredabove.Overtime in soccer

Penalty shootouts in soccer, known as kicks from the penaltymark, which is located 11 m from the goal line, usually occur inknock-out soccer tournaments or cup competitions. After 90 minor extra-time, when the two teams are still tied, they move to the

14 D. Granot, Y. Gerchak / Operations Research Letters 42 (2014) 12–15

penalty shootout stage. As in overtime rules in the NFL, collegefootball, and the CFL, the team to take the first kick is decided by acoin toss. The teams, alternately, attempt up to five penalty kicks. Ifno team is ahead on goals after these five kicks, the teams proceedto the next stage, where they both, alternately, attempt a singlepenalty kick, until one team is ahead on goals and declared thewinner.

In the penalty shootout stage, teams seems to prefer to go first.For example, based on the analysis of 2820 penalty kicks in penaltyshootouts frommajor national and international competitions be-tween 1970 and 2008, it was found in [10] that penalty shootoutsgive an unfair psychological advantage to the team that kicks first,which wins sixty percent of the time, as compared to forty per-cent of the time where the team that kicks second wins. ([7] ar-rives at a different conclusion, and the ‘‘empirical’’ debate maycontinue.) Indeed, in reference to their paper [10], Palacios-Huertasaid: ‘‘Most TV channels cut to the commercial breakwhen the coinis being tossed to decidewhich team takes the first penalty, but ourfindings show that this could be the deciding moment in a drawnmatch’’ [6].

Thus a coin toss to determinewho goes first in penalty shootoutin soccer is unfair for similar reasons to those, say, in college foot-ball or the NFL. An auction, similar to the one proposed above, canbe implemented for penalty shootouts in soccer as follows. Eachof the two teams, A and B, submits a bid, bA and bB, respectively,which designates the distance from the penalty mark, away fromthe goal, from which it is prepared to shoot its penalty kicks if itgoes first. Then, if, say, bA < bB, Team A, that submitted the losingbid, goes second and kicks from the regular penaltymark, i.e., 11mfrom the goal. Team B, that submitted the winning bid, goes first,and attempts its penalty kicks from 11 + αbA + (1 − α)bB metersaway from the goal, for a predeterminedα, 0 ≤ α ≤ 1. Clearly, thelosing team in the bidding process strictly prefers that the winningteam will start further away from the penalty mark, which caststhe auction to determine the team that will go first and second atthe penalty shootout phase in soccer as an auctionwith externalitydiscussed above.

To formally analyze bidding in soccer, consider a symmetricprivate value auction, in which there are two bidders (i.e., teams),who each bids to be the first team to kick the penalty shots. Teams1 and 2 assign values X and Y , respectively, to be the first team thatkicks the penalty shots, which is the maximum they are willing topay, in terms of extra distance from the penalty mark, away fromthe goal line, fromwhich they arewilling to kick the penalty shots ifthey kick first. X and Y are independent and identically distributed(i.i.d.) continuous random variables with a distribution function Fand density function f . Team 1 observes a signal (realization) ofX , and Team 2 observes a signal of Y , but neither team knows thesignal of the other team, aside for the fact that it was a realizationgenerated from F . Both teams are assumed to be risk neutral.

In view of the findings in [10], wewill assume that if both teamskick the penalty shots from the penalty mark, then the team thatkicks first wins the game with a probability 0.6. Further, by defi-nition, if, say, Team 1 observes a signal x of X , then its probabilityof winning if it kicks first from a distance of 11 + x from the goalline and the other team kicks second from the penalty mark, is 0.5.Then, since the probability of Team 1 winning if it kicks first fromthe penalty mark is 0.6, xmust be positive, and accordingly, we as-sume that X and Y are i.i.d. on [c, d], 0 < c < d. Using a linearinterpolation and extrapolation, we will assume that the probabil-ity Team 1 will win, if it kicks first from a distance 11 + z and theother teamkicks second from the penaltymark, is 0.6−

0.1x z, where

z varies from c to d.Consider the mechanism discussed earlier according to which,

if the two teams submit bids b and y, then the team submitting thehigher bid kicks its penalty shots first, from a distance of 11+αb+

(1 − α)y from the goal line, and the other team kicks second fromthe penaltymark. Now, suppose Teams 1 and 2 submit bids of b andy, respectively, and b ≥ y. Then, the probability that Team 1 willwin is P1(b, y) ≡ 0.6−

0.1x (αb+(1−α)y). If Team2places a bid y =

b and kicks first, then, at this point, Team 1’s probability of winningis 1 − (0.6 −

0.1x b) = 0.4 +

0.1x b. For y > b, Team 2 kicks first,

and the probability of Team 1 winning is P2(b, y) ≡ 0.4 +0.1x b +

0.1x (αy + (1 − α)b − b).A strategy for each Team i, i = 1, 2, in the soccer auction is a

function βi : [c, d] → R+, which specifies its bid for every signal.We can assume, for example, that the appropriate sports governingbody, e.g., either FIFA, the governing body in international soccer,or the NFL, in the case of American football, sets the lower andupper bounds, c and d, respectively. Since we assume that the twoteams are symmetric, we will consider only symmetric equilibria,and suppose Team 2 follows the symmetric, increasing anddifferentiable equilibrium strategy β . Thus, if it observes a signal y,it bids β(y). Then, the (expected) utility for Team 1 of winning thepenalty shootouts if it places a bid b, when the utilities of winningand losing the game are normalized to 1 and 0, respectively, is

I(x, b) =

β−1(b)

cP1(b, β(y))f (y)dy +

d

β−1(b)P2(b, β(y))f (y)dy.

Note that P1 and P2 are functions of x, and thus the utility functionI(x, b) is a function of x.

Suppose Team 1 observes a signal x of X , and bids b. We wishto determine the optimal b. For that purpose we seek b whichmaximizes I(x, b). Taking the derivative of I(x, b) with respect tob we obtain: dI(x,b)

db = β−1(b)c

−0.1αx f (y)dy + [0.6 −

0.1x (αb + (1 −

α)β(β−1(b)))] f (β−1(b))β ′(β−1(b))

+ dβ−1(b)

0.1(1−α)

x f (y)dy − [0.4 +0.1bx +

0.1x (αβ(β−1(b)) + (1 − α)b − b)] f (β−1(b))

β ′(β−1(b))= −

0.1x αF(β−1(b)) +

[0.6 −0.1x (αb + (1 − α)β(β−1(b)))] f (β−1(b))

β ′(β−1(b))+

0.1(1−α)

x (1 −

F(β−1(b)))−(0.4+0.1bx +

0.1x (αβ(β−1(b))+(1−α)b−b)) f (β−1(b))

β ′(β−1(b)).

At a symmetric equilibrium, β(x) = b and thus x = β−1(b).Then, a necessary condition for equilibrium is 0 = −

0.1x αF(x) +

(0.6−0.1x β(x)) f (x)

β ′(x) +0.1(1−α)

x (1−F(x))−(0.4+0.1β(x)

x )f (x)β ′(x) , which

simplifies to:

2f (x)(β(x) − x) = β ′(x)(1 − α − F(x)),

where c ≤ x ≤ d.Note that dI(x,b)

db = 0.2 f (β−1(b))β ′(β−1(b))

−0.1x αF(β−1(b)) −

0.1x (αb +

(1 − α)β(β−1(b))) f (β−1(b))β ′(β−1(b))

+0.1(1−α)

x (1 − F(β−1(b))) − ( 0.1bx +

0.1x (αβ(β−1(b))+(1−α)b−b)) f (β−1(b))

β ′(β−1(b)). By definition,β(x) is non-

negative and increasing,which implies that 0.2 f (β−1(b))β ′(β−1(b))

is positive.

Therefore, dI(x,b)db increases in x, at x = β−1(b) it is zero, and thus it

has the same sign as x−β−1(b), or, I(x, b) is concave (see also [3]).Thus, we conclude that if β(x) is non-negative and increasing, theabove necessary conditions are also sufficient.

We are able to draw several interesting conclusions from thenecessary and sufficient conditions for equilibrium. First note thattruthful bidding (β(x) = x) would occur only if 1 − α − F(x) =

0. Therefore, since for a positive f (x) on its domain, 1 − α −

F(x) = 0 only for a set of measure zero, we can conclude that fora non-negative, symmetric, increasing and differentiable equilib-rium strategy β:

Proposition 1. With probability 1, any implementation of a convexcombination of first-price and second-price auction mechanism in thepenalty shootout stage in soccer will not induce truthful bidding.

D. Granot, Y. Gerchak / Operations Research Letters 42 (2014) 12–15 15

Proposition 2. For an implementation of a convex combination offirst-price and second-price auctionmechanismwith a givenα, signalsx for which F(x) > 1− α will induce under-bidding, and signals x forwhich F(x) < 1 − α will induce over-bidding.

In particular, since 0 < F(x) < 1 with probability 1, wecan similarly derive from the necessary and sufficient conditionsfor equilibrium that for a first-price (resp., second-price) auctionmechanism (i.e., α = 1 (resp., α = 0)):

Proposition 3. An implementation of a first-price auction mecha-nism in the penalty shootout stage in soccer will always induce under-bidding of the teams’ true valuations.

Proposition 4. An implementation of a second-price auction mecha-nism in the penalty shootout stage in soccer will always induce over-bidding of the teams’ true valuations.

For 0 < α < 1, (1 − α − F(x)) = 0 in the interior of (c, d).Therefore, in order to solve the first order differential equation,2f (x)(β(x) − x) = β ′(x)(1 − α − F(x)), for a given α, we dividethe domain, (c, d), into two regions: {x : c < x < F−1(1− α)} and{x : F−1(1 − α) < x < d}. A general solution for β(x) is: β(x) = xg u(t)q(t)dt−E

u(x) , where u(t) = ep(t)dt , p(t) =

−2f (t)1−α−F(t) , q(t) =

−2f (t)t1−α−F(t) , g is the lower limit, either c or F−1(1− α), depending onthe region, E is the integration constant, whose value may dependon the region, and x varies in the appropriate region. In our case,

the expression for β(x) simplifies to β(x) = x −

xg (1−α−F(t))2dt−E

(1−α−F(x))2,

and one can easily verify that β(x) indeed satisfies the necessarycondition 2f (x)(β(x) − x) = β ′(x)(1 − α − F(x)) in the two re-gions. Now, β(x) has to be non-negative and bounded (e.g., by thelength of the soccer field) for all signals x. Thus, the constant de-rived from the evaluation of the integral at its lower limit, g , plusthe integration constant, E, must equal zero, and the expression forβ(x) simplifies to β(x) = x −

(1−α−F(x))2dx(1−α−F(x))2

. Further, for the sameboundedness and non-negativity reasons,

(1 − α − F(x))2dx has

to converge faster to zero than (1−α−F(x))2 when 1−α−F(x) ap-proaches zero, which is satisfied, e.g., for the uniform distribution.

Specifically, for the uniform distribution, f (x) =1

d−c , 0 < c ≤

x ≤ d, and 1−α −x−cd−c = 0, β(x) =

2x3 +

c3 +

(1−α)(d−c)3 , which in-

creases in x, and, again, one can easily verify thatβ(x) indeed solvesthe differential equation 2f (x)(β(x) − x) = β ′(x)(1 − α − F(x)).For a first-price auction, β(x) =

2x3 +

c3 ≤ x, and for a second-price

auction, β(x) =2x3 +

d3 ≥ x.

Finally, recall that for α such that 1− α −x−cd−c = 0, i.e., for α =

α∗≡

d−xd−c , we can conclude directly from the differential equation

that β(x) = x. However, β(x) =2x3 +

c3 +

(1−α)(d−c)3 , evaluated at

α∗, also coincides with x. That is, the solution derived for β(x), forthe case when α = α∗, is also valid for α∗.

Naturally, all the results derived above are also valid for footballand chess. In fact, unlike the penalty shootout stage in soccer,the team which starts first on offense in the overtime period inthe NFL, or the player who plays the black pieces for a drawin the Armageddon stage in a chess competition, has more thana psychological advantage over the opposing team or player,respectively.

Acknowledgments

The authors are grateful tomany useful suggestionsmade by ananonymous referee which led to a greatly improved presentationof this note. The first author is grateful to Ilan Adler for introduc-ing him to the Armageddon stage in chess, and the second authorwishes to thank Steve Brams for sharing with him the idea of bid-ding in sports. Research was supported by a Canada Natural Sci-ences and Engineering Research Council grant.

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