Team Tennis competitions produce aggregate scores for teams, and thus team rankings, based on head-to-head matchups of individual team members. Similar scoring rules can be used to rank any two groups that must be compared on the basis of paired elements. We explore such rules in terms of their strategic and social choice characteristics, with particular emphasis on the role of cycles. We first show that cycles play an important role in promoting competitive balance, and show that cycles allow for a maximum range of competitive balance within a league of competing teams. We also illustrate the impact that strategic behavior can have on the unpredictability of competition outcomes, and show for a general class of team tennis scoring rules that a rule is strategy-proof if and only if it is acyclic (dictatorial) and manipulable otherwise. Given the benefits of cycles and their relationship with manipulability, a league valuing competitive balance may invite such social choice violations when choosing a scoring rule.

Tennis is typically viewed as an individual sport. At school, club, and professional levels, however, individual matchups are aggregated into a team result. A National Collegiate Athletic Association (NCAA) team tennis competition, for example, features six individual matches and either one or three doubles matches.

More generally, what we refer to as

In the present paper we examine the social choice characteristics of team tennis scoring rules as they relate to strategic manipulability and ambiguity. In particular we are interested in the role of

For a general class of team tennis scoring rules, we show that a rule is strategy-proof if and only if it is dictatorial, meaning only one of the element pairs matters for the entire group ranking. This in turn means that a rule is strategy-proof if and only if it is acyclic, and manipulable whenever it allows cycles. Though such results are often portrayed as negative in the social choice literature, we argue that this is not the case in the context of team tennis, since we show that both of the noted social choice violations promote competitive balance between opposing teams. In fact, we demonstrate that a full range of competitive balance, from cycles between the absolute best and worst teams in a league to perfect balance, can be achieved with a family of team tennis scoring rules that we call weighted-sum team tennis scoring rules.

Unlike traditional arenas of social choice analysis, third-party (spectator) welfare is paramount in sport. Following the seminal contributions of Neale [

A few previous studies have examined the social choice characteristics of team sport. Avgerinou [

After providing some basic results regarding cycles and competitive balance and the uncertainty of outcome in the second section, we provide our general team tennis model and establish some results on team tennis scoring and cycles in the third section. Then in the fourth section we follow Boudreau et al. [

We begin with just a few of the most basic aspects of the model and will build upon them further in the next section. Consider a league of

However the scoring rule aggregates individual match outcomes, we say that the rule is

We can now establish a very simple proposition, adapted from Boudreau and Sanders [

Given an acyclic scoring rule, after a full schedule it must be the case that one team has

We also develop a proposition regarding match uncertainty of outcome, which will be explained more fully after the statement of the proposition itself.

In a league consisting of

Whether the ranking methodology is cyclic or acyclic, the first two match outcomes are a priori uncertain. As the outcomes of the first two matches could never generate a cycle, there are no restrictions on the outcomes of the first two matches. Under the acyclic rule, the third match outcome is a priori known whenever the alternative third match outcome would generate a cycle. Under the cyclic rule, the third match outcome is never known a priori. □

We now flesh out the model. Consider a match between teams

We assume that the rank ordering of players for each team is given, fixed, and transitive, perhaps decided by the team’s coach or by prior intra-squad competition.

We define a team tennis match as a series of events in which each competitive element of

In a team tennis dual meet between

Standard team tennis scoring is a case of

We have already established that cyclical scoring rules improve competitive balance and uncertainty of outcome in leagues, so we now investigate the degree to which weighted-sum team tennis rules allow for cycles. To begin we use standard rule of

By assumption, the following individual player quality rankings obtain for any three teams,

More succinctly, player quality ordering across the three teams can be represented as

In Example 1, one notes immediately the absence of a Condorcet winner in the set of team rankings in Equation 2a-c. When

Our next example shows that cycles among teams with a maximum amount of disparity between their win differentials, or between any teams, within a league are always possible with weighted-sum team tennis scoring rules, so long as they are not dictatorial. We say that a team tennis scoring rule is

To work with a general, non-dictatorial weighted-sum team tennis scoring rule, we must also define the index

Consider a league of

Based on the definition of

We also note that the example in the proof above can be augmented to illustrate cycles between teams with any number of wins by simply expanding or contracting the number of teams within the template ordering above, or by including “dummy” teams whose

Our next result shows that perfect competitive balance is possible for a league using a non-dictatorial weighted-sum team tennis scoring rule. A state of perfect balance is one in which all teams have the same win total. To show this, we must define another threshold index,

Suppose a league’s elements rank as follows:

From the definition of

Each team therefore has exactly

The last result major of this section sets up the rest of the paper by relating the presence of cycles and dictatorship to the prospect of strategic behavior in team tennis. Rather than truthfully listing the ranking of their players, some tennis coaches may instead choose to list their stronger players lower in the order in order to give them easier victories, sacrificing one match in order to win others. A similar situation, dubbed the “tennis coach problem”, was considered by Arad [

As mentioned in the introduction, we consider a scoring rule to be strategy-proof if it is a (weakly) dominant strategy for a team to rank its players truthfully. We assume that it is each team’s priority to win a dual meet, and that they will misrepresent their player ranking if doing so will help them to that end.

For the “if” part of the statement, note that when

For the “only if” part of the statement, suppose

Based on that result, in particular, the example used in the proof, and the fact that we assume all players must be strictly ranked overall, so a dictatorial rule must be acyclic, we have the following.

We know that cycles are possible for team tennis scoring, and that they are linked to interesting properties of leagues. But just because they are possible does not mean they are likely. To determine the proportion of ranking sequences that generate a cycle under standard scoring, we constructed a Java program to search the substantial space of possible rankings across three teams, each with three components, in the case that each team engages in a dual meet against each of the other two teams. In the simplified case of three teams and three individuals per team, there are

Strategic voting occurs when individuals have an incentive to falsely represent their preferences. Gibbard [

Consider two teams,

From

Note that

In

For each representation strategy employed by

This mixed strategy Nash equilibrium can be described as a pair of evenly mixed strategies. Across the two teams, each representation strategy is chosen (independently) with likelihood

As

In 12 of the 20 true orderings, there is no incentive to misrepresent one’s intra-squad player rankings and thus no representation game. These include orderings Equation (8a–e,h,k–o,r). In each such case, there is a dominant team that wins regardless of the strategy profile that would obtain in a representation game. If one of true orderings Equation (8f,g,i,j,p,q,s), or (8t) obtains, a representation game ensues. By the same reasoning as was applied in the previous analysis of ordering Equation (8f), the Nash equilibrium for each of these eight true orderings is found to be an evenly distributed mixed strategy (i.e., each representation chosen with probability

In 8 of 20 possible player quality orderings, the representation game occurs and allows a team that would never win in the absence of strategy to win with positive probability. We therefore conclude that representation strategy, like ranking cycles, is expected to increase competitive balance and

Under the uncertainty of outcome hypothesis, we expect representation strategy to increase fan welfare in team tennis. Moreover, strategic ranking obtains over a much larger set of orderings than do ranking cycles. As in the section concerning ranking cycles, consider a set of three teams, each of which features three individual players. As was found previously, there are 1680 possible player orderings across three such teams, and 30 orderings (1.79 percent) result in a ranking cycle. We now examine the likelihood that strategic ranking behavior occurs in all three team matches between

In the rank-ordering

In what many believe to be the earliest journal article in sports economics (see, e.g., Sanderson and Siegfried [

Although the absolute quality of play influences demand and absolute investments in training are socially efficient (Lazear and Rosen [

Schmidt and Berri [

The seminal work of El-Hodiri and Quirk [

To understand the uncertainty of outcome hypothesis, one might consider three types of fans in a dual match: team A supporters, team B supporters, and neutral fans. Neutral fans may consist of bettors and of those who simply like to observe a new or uncertain event. Under the uncertainty of outcome hypothesis, neutral fans are expected to lose interest as the match becomes more certain in favor of one team. It becomes more difficult to establish gambles (i.e., simplistic betting markets cease and a handicap betting market must be designed), and neutral fans who do not gamble may feel that the match outcome is a foregone conclusion.

In

Shane Sanders created some of the primary theoretical results for the paper. Justin Ehrlich solely rendered all computational results. James Boudreau developed other primary theoretical results and revised the original theoretical section.

The authors declare no conflict of interest.

The NCAA division a team is in dictates whether it plays one or three doubles matches. This is discussed a bit more in

The assumption that

For simplicity, we abstract from the reality that individual players and doubles pairs are ranked separately.

In reality, opposing NCAA team tennis coaches do submit intra-squad rankings simultaneously just before a match. The rankings cannot be edited upon submission.

John Isner and Nicolas Mahut met in a first round match of the 2010 Wimbeldon Championships. Isner was victorious in the longest professional tennis match ever played (11 h and 5 min of play over parts of three days; 183 games). The two players essentially traded 136 games in the fifth set before Isner won the 137th and 138th games of the set.

Arad [

This calculation assumes that player orderings within team are known and invariant.

Numerous betting scandals involve betting fans seeking information from athletes and coaches prior to a match.

Several economists have credited Rottenberg [

Handicap betting markets, in which a “handicapper” or “market-maker” attempts to establish a fair bet through artificial scoring compensation or other means, are commonly thought to be more manipulable than traditional betting markets. For example, the “market-maker” may have an incentive to under-compensate or over-compensate a team. Moreover, a player or coach can sometimes alter the outcome of a handicapped bet without altering the overall match outcome. In college basketball, several point-shaving scandals have arisen from handicapped betting markets. Such problems are explored in, e.g., Wolfers [

All potential individual matches involving

1 | 1 | 1 | |

0 | 1 | 1 | |

0 | 0 | 1 |

Possible misrepresentations of player rank-ordering within team x.

Position | True Rank | Chosen Rank A | Chosen Rank B | Chosen Rank C | Chosen Rank D | Chosen Rank E | Chosen Rank F |
---|---|---|---|---|---|---|---|

1 | |||||||

2 | |||||||

3 |

Normal Form of a representation game between x and y.

A | B | C | D | E | F | ||
---|---|---|---|---|---|---|---|

A | (1,0) | (1,0) | (1,0) | (1,0) | (0,1) | (1,0) | |

B | (1,0) | (1,0) | (1,0) | (1,0) | (1,0) | (0,1) | |

C | (1,0) | (0,1) | (1,0) | (1,0) | (1,0) | (1,0) | |

D | (0,1) | (1,0) | (1,0) | (1,0) | (1,0) | (1,0) | |

E | (1,0) | (1,0) | (1,0) | (0,1) | (1,0) | (1,0) | |

F | (1,0) | (1,0) | (0,1) | (1,0) | (1,0) | (1,0) |

Representation Game Nash equilibrium outcomes.

Ordering | Outcome without Strategy | Outcome with Strategy |
---|---|---|

(8f) | ||

(8g) | ||

(8i) | ||

(8j) | ||

(8p) | ||

(8q) | ||

(8s) | ||

(8t) |