Round‐Robin Tournaments with a Dominant Player

Published date01 October 2017
AuthorAlex Krumer,Reut Megidish,Aner Sela
Date01 October 2017
DOIhttp://doi.org/10.1111/sjoe.12204
©The editors of The Scandinavian Journal of Economics 2016.
Scand. J. of Economics 119(4), 1167–1200, 2017
DOI: 10.1111/sjoe.12204
Round-Robin Tournaments with a Dominant
Player
Alex Krumer
Swiss Institute for Empirical Economic Research (SEW), University of St Gallen,
CH-9000 St Gallen, Switzerland
alexander.krumer@unisg.ch
Reut Megidish
Sapir Academic College, M.P. Hof Ashkelon 79165, Israel
reutmeg@gmail.com
Aner Sela
Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
anersela@bgumail.bgu.ac.il
Abstract
We analyze the subgame perfect equilibrium of the round-robin tournament with one
strong (dominant) and two weak players, and we compare this tour nament and the
one-stage contest with respect to the players’ expected payoffs, expected total effort,
and their probabilities of winning. We find that if the contest designer’s goal is to
maximize the players’ expected total effor t, then – if the asymmetry between the players
is relatively low – the one-stage contest should be used. However, if the asymmetry
is relatively high, then the round-robin tournament should be used.
Keywords: All-pay auctions; multistage contests
JEL classification:D44; O31
I. Introduction
Tournaments are prevalent in many areas of life, including labor markets
(Lazear and Rosen, 1981; Prendergast, 1999), political races (Klumpp and
Polborn, 2006), R&D (Harris and Vickers, 1985; Zizzo, 2002; Breitmoser
et al., 2010), rent seeking (Tullock, 1980; Gradstein and Konrad, 1999),
and sports (Rosen, 1986; Szymanski, 2003; Harbaugh and Klumpp, 2005).
In this paper, we focus on the round-robin tournaments in which every
player or team competes against all the others, and in every stage a
player plays a pairwise match against a different opponent. Such particular
tournaments commonly take place in professional football and basketball
leagues, but they can sometimes be seen in other domains. To illustrate,
in the 2015 elections for Israel’s Knesset, a representative of each party
1168 Round-robin tournaments with a dominant player
was invited for a televised debate. This was organized as a round-robin
tournament where, at each stage, the representatives of the parties were
divided into different pairs, with each pair confronting each other for
several minutes.
In the literature on contests, the most common goal, especially in
sport contests, is to maximize the players’ total effor t (Szymanski, 2003).
However, the contest designer might also want to increase the competitive
balance by decreasing the differences among the players’ probabilities of
winning; alternatively, the contest designer might want to affect the identity
of the winner by determining the players’ probabilities of winning (Groh
et al., 2012). This can be done by choosing the type of tournament.
We address these issues by comparing the round-robin tournament and
the standard one-stage contest, in which all the players compete against
each other only once in one grand contest. The comparison is made with
respect to the players’ expected payoffs, their probabilities of winning, and
their expected total effort. It is important to note that when the players
are asymmetric, the results of the round-robin tournament depend on the
allocations of players in the different stages of the tournament. Thus,
because the number of different allocations grows exponentially with the
number of players, we focus on the simple case of three players where one
is dominant (i.e., one player has a higher value of winning than the other
(weaker) players). Over the years, this for mat of round-robin tournaments
with three players has been used in many Olympic Games tournaments,
such as wrestling, badminton, women’s soccer, etc. It has also been used
in several geographical zones of the FIFA World Cup qualifications. In our
round-robin tournament, there are three possible allocations of the players,
all of which are considered in this paper. For both types of contests, each
match is modeled as an all-pay contest.1In the all-pay contest (auction),
the contestant with the highest effort (output) wins the contest, but all
the contestants bear the cost of their effort.2
We find that, independent of the allocation of players, the expected
payoffs of the weak players in the round-robin tournament are higher than
or equal to their expected payoffs in the one-stage contest. In contrast,
depending on the players’ allocation in the round-robin tournament, the
expected payoff of the dominant player in the one-stage contest can be
either higher or lower than in the round-robin tournament. Furthermore, if
the dominant player is allocated in the first and the last stages of the round-
1Numerous applications of the all-pay contest have been made to rent-seeking and lobbying in
organizations, R&D races, political contests, promotions in labor markets, trade wars, military
and biological wars of attrition (e.g., Che and Gale, 1998; Moldovanu and Sela, 2001).
2The all-pay contest is the limit point of the popular Tullock contest with the success function
pi(x1,x2)=(xi)r/[(x1)r+(x2)r], i=1, 2, when rconverges to infinity. Thus, we can conjecture
that our results hold for the Tullock success function when ris sufficiently large.
©The editors of The Scandinavian Journal of Economics 2016.
A. Krumer, R. Megidish, and A. Sela 1169
robin tournament, then his expected payoff in the round-robin tournament
is higher than in the one-stage contest. The intuitive explanation for this
is that if the dominant player wins in the first stage, then his expected
value of winning increases and his opponents’ expected values of winning
decrease in the following stages. In addition, by playing in the last stage,
the dominant player might compete against an opponent who lost in the
previous stages, and who therefore has a low chance of winning the
tournament. This situation enables the dominant player to win in the last
stage and, particularly, to win the entire tournament without exerting much
effort. Therefore, allocation in the first and last stages is favorable for the
dominant player, who, in that case, prefers the round-robin tournament
over the one-stage contest.
However, usually, players cannot choose the stages in which they are
allocated, and actually the allocation of players in the round-robin
tournament is randomly determined (i.e., each possible allocation of players
is chosen by the same probability). In that case, the dominant player’s
expected payoff in the round-robin tournament will be higher than in
the one-stage contest, given that the asymmetry between the players is
relatively low, and vice versa if the asymmetry is high. Thus, while the
weak players prefer the round-robin tournament, the dominant player does
not necessarily prefer either of the contests.
Using this analysis of the subgame perfect equilibrium, we then calculate
the dominant player’s probability of winning in the round-robin tournament.
Then, we show by numerical analysis that if the players are randomly
allocated in the round-robin tournament, independent of the asymmetry
of the players, the dominant player’s probability of winning in the one-
stage contest is higher than in the round-robin tournament when the players
are randomlyallocated. These interesting findings indicate that the common
intuition – according to which the dominant player’s probability of winning
is always higher in the round-robin tournament than in the one-stage
contest – is not correct. However, we also show that, independent of the
level of asymmetry, the dominant player’s probability of winning is highest
when the weak players are matched in the second stage. The reason for
this is again that if the dominant player wins in the first stage, then he
dramatically increases the difference in his expected value of winning and
his opponents’ values of winning in the next stages, and accordingly he
increases his probability to win his next game. However, if the dominant
player does not play in the first stage, he might play against a weak player
who already won in the previous stage and who will therefore have an
expected value that is higher than his own (i.e., the dominant player will
no longer be dominant). Thus, if a contest designer wishes to maximize
the dominant player’s probability of winning, then he should organize a
©The editors of The Scandinavian Journal of Economics 2016.

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