Stochastic Rationality and Team Response Functions

GAMES PLAYED BY TEAMS OF PLAYERS

2.5 Stochastic Rationality and Team Response Functions

a scoring rule but satisfies the Nash convergence property. Finally, we conjecture that the class is much broader than scoring rules, including many other anonymous and neutral collective choice rules. Scoring rules operate only on the individual ordinalrankings of estimated expected payoffs. One imagines that there are many collective choice rules that operate on the cardinal values of the estimates and also have the Nash convergence property, such as weighted average rules.

response functions to satisfy these two properties. We first show that payoff mono- tonicity of team response functions requires only two weak assumptions on the collective choice rule, unanimity and positive responsiveness. On the other hand, rank dependence holds only for a more restricted class of neutral collective choice rules. Many non-neutral collective choice rules, such as those that give a status quo advantage to an action, will fail to satisfy rank dependence, as the last example in section 3.1 demonstrates. Second, we show that rank dependence is satisfied for 𝐾^𝑡 =2 with any collective choice rule satisfying unanimity, positive responsiveness, and neutrality, and for𝐾^𝑡 > 2 with plurality rule or weighted average rules.

Payoff Monotonicity For any team game, Γ, 𝑃^𝐶

𝑡 depends on the strategy profile 𝛼, the distribution of member’s estimation errors, 𝐹^𝑡, and the team collective choice rule,𝐶^𝑡. In this section we identify conditions on𝐶^𝑡that are sufficient for𝑃^𝐶

𝑡 to be payoff monotone for all admissible𝐹^𝑡. The formal definition of payoff monotonicity is given below.

Definition 4. A team collective choice rule𝐶^𝑡 satisfiesPayoff Monotonicityif, for all𝑎^𝑡

𝑘, 𝛼, 𝛼^′: 𝑈^𝑡

𝑘(𝛼) > 𝑈^𝑡

𝑘(𝛼^′)and𝑈^𝑡

𝑙(𝛼) =𝑈^𝑡

𝑙(𝛼^′) ∀𝑙 ≠ 𝑘 ⇒ 𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡(𝛼)) > 𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡(𝛼^′)).

Specifically, we require team collective choice rules to satisfy two axioms: unanimity and positive responsiveness. The first condition, unanimity, simply states that if all members of the team estimate that 𝑎^𝑡

𝑘 has the highest expected utility, then it is uniquely chosen by𝐶^𝑡.17

Definition 5. A team collective choice rule𝐶^𝑡 satisfiesUnanimityif:

𝑈b^𝑡

𝑖 𝑘 >𝑈b^𝑡

𝑖𝑙 for all𝑖 ∈𝑡and for all𝑙 ≠ 𝑘 ⇒𝐶^𝑡(𝑈b^𝑡) ={𝑎^𝑡

𝑘}.

In addition to using this axiom to prove payoff monotonicity, it also guarantees that team response functions are interior, in the sense that every action is chosen with positive probability. The second axiom, positive responsiveness, requires that the team choice responds positively to all members of a team increasing their estimated expected payoff of an action, keeping all other estimated expected payoffs the same.

The following definition is used in the statement of the axiom.

Definition 6. A profile 𝑈˜^𝑡 of member estimated expected utilities isa monotonic transformation of𝑈ˆ^𝑡 with respect to action 𝑎^𝑡

𝑘 if, for all members𝑖 ∈ 𝑡, we have

˜ 𝑈^𝑡

𝑖 𝑘 ≥𝑈ˆ^𝑡

𝑖 𝑘 and𝑈˜^𝑡

𝑖𝑙 =𝑈ˆ^𝑡

𝑖𝑙 for all𝑙 ≠ 𝑘.

17For the "standard" case of games played by one-person teams, unanimity implies that if𝑛=1 then every team equilibrium is equivalent to a quantal response equilibrium of the strategic form game,[𝑇 , 𝐴, 𝑢].

Definition 7. A team collective choice rule 𝐶^𝑡 satisfies Positive Responsiveness if 𝑎^𝑡

𝑘 ∈ 𝐶^𝑡(𝑈ˆ^𝑡) ⇒ 𝑎^𝑡

𝑘 ∈ 𝐶^𝑡(𝑈˜^𝑡) ⊆ 𝐶^𝑡(𝑈ˆ^𝑡), for all 𝑎^𝑡

𝑘, 𝑈ˆ^𝑡 and all monotonic transformations𝑈˜^𝑡of𝑈ˆ^𝑡 with respect to action𝑎^𝑡

𝑘.

This definition of Positive Responsiveness is essentially a cardinal version of the usual definition of positive responsiveness from the social choice literature. It says that if an action 𝑎^𝑡

𝑘 is chosen at some profile of estimated expected utilities, and all team members’ estimates of the expected utility of that action weakly increase, ceteris paribus, then𝑎^𝑡

𝑘 must still be chosen, and no new actions can be added to the choice set.

Many collective choice rules satisfy positive responsiveness. For example, any weighted average rule, where the team choice corresponds to the action with the highest weighted average of individual members estimates, is positively responsive.

Plurality rule also clearly satisfies this condition. In this section we consider a class of collective choice rules, calledgeneralized scoring rules, and show that positive responsiveness is satisfied for any such collective choice rule. A generalized scoring rule is substantially more general than the standard definition of a scoring rule in the social choice literature, which was defined in the previous section as ananonymous scoring rules (i.e., all the individual scoring functions are the same). Generalized scoring rules relax the anonymity requirement that all individual scoring functions are the same. It includes a wide range of non-anonymous collective choice rules, including dictatorial rules.

Definition 8. A team collective choice rule𝐶^𝑡is aGeneralized Scoring Ruleif there exists a profile of individual scoring functions,(𝑆^𝑡

1, ...𝑆^𝑡

𝑛^𝑡), such that for all𝑎^𝑡

𝑘 ∈ 𝐴^𝑡 and for all𝑈ˆ^𝑡 ∈ ℜ^{𝐾𝑡𝑛𝑡}, 𝑎^𝑡

𝑘 ∈𝐶^𝑡(𝑈ˆ^𝑡) if and only ifÍ𝑛^𝑡 𝑖=1𝑆^𝑡

𝑖 𝑘(𝑈ˆ^𝑡

𝑖) ≥ Í𝑛^𝑡 𝑖=1𝑆^𝑡

𝑖𝑙(𝑈ˆ^𝑡

𝑖) for all𝑙 ≠ 𝑘.

Proposition 1. All generalized scoring rules satisfy positive responsiveness.18 Proof. Positive responsiveness follows from the fact that the value of the team score function evaluated at any alternative is weakly increasing in that alternative’s estimated expected utility for each team member, and weakly decreasing in every

other alternative’s estimated expected utility. □

We can now state the main result of this subsection.

18If𝑠^𝑡

𝑖1> 𝑠^𝑡

𝑖2for all𝑖∈𝑡, then the scoring rule also satisfies unanimity.

Theorem 3. If𝐹^𝑡 is admissible and𝐶^𝑡 satisfies unanimity and positive responsive- ness then𝑃^𝐶

𝑡

satisfies payoff monotonicity.

Proof. Let 𝐶^𝑡 satisfy positive responsiveness and unanimity and 𝐹^𝑡 admissible.

Suppose that𝑈^𝑡

𝑘−𝑈

′𝑡

𝑘 =𝛿 >0, and𝑈^𝑡

𝑙 =𝑈

′𝑡

𝑙 ,∀𝑙 ≠ 𝑘. Then for all realizations of the estimation errors𝜖^𝑡, we have that𝑈^𝑡+𝜖^𝑡is a monotonic transformation of𝑈

′𝑡

+𝜖^𝑡with respect to𝑎^𝑡

𝑘. So by positive responsiveness of𝐶^𝑡 we have that if𝑎^𝑡

𝑘 ∈𝐶^𝑡(𝑈

′𝑡

+𝜖^𝑡) then 𝑎^𝑡

𝑘 ∈ 𝐶^𝑡(𝑈^𝑡 + 𝜖^𝑡), and if 𝑎^𝑡

𝑙 ∈ 𝐶^𝑡(𝑈^𝑡 + 𝜖^𝑡) then 𝑎^𝑡

𝑙 ∈ 𝐶^𝑡(𝑈

′𝑡

+ 𝜖^𝑡). So 𝑔^𝐶

𝑡

𝑘 (𝑈^𝑡+𝜖^𝑡) ≥ 𝑔^𝐶

𝑡

𝑘 (𝑈

′𝑡+𝜖^𝑡)for all𝜖^𝑡, and therefore𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) ≥ 𝑃^𝐶

𝑡

𝑘 (𝑈

′𝑡). To show the strict inequality,𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) > 𝑃^𝐶

𝑡

𝑘 (𝑈

′𝑡), we show that there exists a region𝛽 ⊂ ℜ^{𝐾𝑡×𝑁𝑡} with positive measure such that if 𝜖^𝑡 ∈ 𝛽, then 𝑔^𝐶

𝑡

𝑘 (𝑈^𝑡 +𝜖^𝑡) > 𝑔^𝐶

𝑡

𝑘 (𝑈

′𝑡 +𝜖^𝑡). In particular, unanimity of𝐶^𝑡is used as follows to construct𝛽such that if𝜖^𝑡 ∈ 𝛽, then 𝑔^𝐶

𝑡

𝑘 (𝑈^𝑡 +𝜖^𝑡) = 1 > 𝑔^𝐶

𝑡

𝑘 (𝑈

′𝑡

+𝜖^𝑡) = 0. That is, such that 𝑎^𝑡

𝑘 is uniquely chosen under𝑈^𝑡+𝜖^𝑡, and not chosen under𝑈

′𝑡

+𝜖^𝑡. Let ˜𝑈^𝑡be an estimated expected utility profile such that all team members strictly prefer some action 𝑎^𝑡

𝑙 to action 𝑎^𝑡

𝑘, all members prefer 𝑎^𝑡

𝑘 to all other actions𝑎^𝑡

𝑚 (i.e., all members rank 𝑎^𝑡

𝑘 second), and for all members we have ˜𝑈^𝑡

𝑙 −𝑈˜^𝑡

𝑘 = ^𝛿₂. Define:

𝛽 ={𝑈˜^𝑡−𝑈

′𝑡

+𝜉 :𝜉_𝑘 ∈ (0, 𝛿

4), 𝜉_𝑙 ∈ (−𝛿

4,0), 𝜉_𝑚 < 0} Then if𝜖^𝑡 ∈ 𝛽, by unanimity we have𝐶^𝑡(𝑈

′𝑡

+𝜖^𝑡) = {𝑎^𝑡

𝑙} and𝐶^𝑡(𝑈^𝑡 +𝜖^𝑡) ={𝑎^𝑡

𝑘}, so𝑔^𝐶

𝑡

𝑘 (𝑈^𝑡+𝜖^𝑡) = 1 > 𝑔^𝐶

𝑡

𝑘 (𝑈

′𝑡

+𝜖^𝑡) = 0. 𝛽 is an open set and hence has positive measure since the distribution of𝜖^𝑡has full support. Therefore𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) > 𝑃^𝐶

𝑡

𝑘 (𝑈

′𝑡

),

as desired. □

Rank Dependence

In this section we show that 𝑃^𝐶

𝑡 satisfies rank dependence for 𝐾^𝑡 = 2 with any collective choice rule satisfying unanimity, positive responsiveness and neutrality, and for𝐾^𝑡 > 2 with plurality rule and weighted average rules. The formal definition of rank dependence is:

Definition 9. A team collective choice rule𝐶^𝑡satisfiesRank Dependenceif, for all 𝑎^𝑡

𝑘

, 𝑎^𝑡

𝑙

, 𝛼, 𝑈^𝑡

𝑘(𝛼) > 𝑈^𝑡

𝑙(𝛼) ⇒ 𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡(𝛼)) > 𝑃^𝐶

𝑡

𝑙 (𝑈^𝑡(𝛼))

Neutrality is an essential property for proving that team response functions satisfy rank dependence. Informally a neutral team collective choice rule is one that is not biased against or in favor of any particular action. This is analogous to the neutrality axiom from the social choice literature.

Let 𝜓 : 𝐴^𝑡 → 𝐴^𝑡 be any permutation of team actions. Denote by 𝑈^{𝑡 ,𝜓} = (𝑈^𝑡

𝜓(1), ..., 𝑈^𝑡

𝜓(𝐾^𝑡))the permuted profile of expected utilities and by ˆ𝑈^{𝑡 ,𝜓} ≡ (𝑈^𝑡

𝜓(1)+ 𝜖^𝑡

𝜓(1), ..., 𝑈^𝑡

𝜓(𝐾^𝑡) +𝜖^𝑡

𝜓(𝐾^𝑡)) the permuted profile of estimated expected utilities. We can then define neutrality formally.

Definition 10. A team collective choice rule𝐶^𝑡satisfiesNeutralityif, for all𝑎^𝑡

𝑘,𝑈ˆ^𝑡, for all permutations𝜓,𝑎^𝑡

𝑘 ∈𝐶^𝑡(𝑈ˆ^𝑡) ⇔𝑎^𝑡

𝜓(𝑘) ∈𝐶^𝑡(𝑈ˆ^{𝑡 ,𝜓})

Neutrality, along with admissibility of𝐹^𝑡, imply that when the expected payoffs of team actions are permuted, the team choice probabilities are permuted.

Lemma 1. If𝐹^𝑡is admissible and𝐶^𝑡satisfies neutrality, then𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) =𝑃^𝐶

𝑡

𝜓(𝑘)(𝑈^{𝑡 ,𝜓}) for all expected utility profiles,𝑈^𝑡, actions,𝑎^𝑡

𝑘, and permutations𝜓.

Proof. By neutrality of𝐶^𝑡, for any expected utility profile, 𝑈^𝑡, action, 𝑎^𝑡

𝑘, belief error profile,𝜖^𝑡, and permutation 𝜓, 𝑔^𝐶

𝑡

𝑘 (𝑈ˆ^𝑡) = 𝑔^𝐶

𝑡

𝜓(𝑘)(𝑈ˆ^{𝑡 ,𝜓}), that is the probability that 𝑎^𝑡

𝑘 is chosen at ˆ𝑈^𝑡 is equal to the probability that 𝑎^𝑡

𝜓(𝑘) is chosen at ˆ𝑈^{𝑡 ,𝜓}. Therefore 𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) = ∫

𝜖^𝑡

𝑔^𝐶

𝑡

𝑘 (𝑈ˆ^𝑡)𝑑𝐹^𝑡(𝜖^𝑡) = ∫

𝜖^𝑡

𝑔^𝐶

𝑡

𝜓(𝑘)(𝑈ˆ^{𝑡 ,𝜓})𝑑𝐹^𝑡(𝜖^𝑡). Finally, since the estimation errors are i.i.d, ∫

𝜖^𝑡

𝑔^𝐶

𝑡

𝜓(𝑘)(𝑈ˆ^{𝑡 ,𝜓})𝑑𝐹^𝑡(𝜖^𝑡) = ∫

𝜖^𝑡

𝑔^𝐶

𝑡

𝜓(𝑘)(𝑈ˆ^{𝑡 ,𝜓})𝑑𝐹^𝑡(𝜖^{𝑡 ,𝜓}) = 𝑃^𝐶

𝑡

𝜓(𝑘)(𝑈^{𝑡 ,𝜓}). □

A corollary to the lemma is that when two actions have equal expected payoffs, the team must play these actions with equal probability. It is easy to see that non-neutral collective choice rules can lead to violations of rank dependence. For example, collective choice rules that favor one action (e.g. a status quo action) over another will generally lead to violations, as in the last example of Section 3.1 with a 2/3 voting rule. Consider𝐾^𝑡 =2 and a choice rule that selects action𝑎^𝑡

1if and only if all team members estimate its expected utility to be greater than that of action𝑎^𝑡

2, and selects action𝑎^𝑡

2 otherwise. For any admissible 𝐹^𝑡, if the size of the team is large enough, a team using this choice rule will select action 𝑎^𝑡

2more often than action 𝑎^𝑡

1even when𝑈^𝑡

1(𝛼) > 𝑈^𝑡

2(𝛼).

Next, for the case of 𝐾^𝑡 = 2, we prove that neutrality, together with unanimity and positive responsiveness is sufficient to guarantee that a team response function satisfies rank dependence for all admissible𝐹^𝑡. This is proved below.

Theorem 4. If𝐾^𝑡 =2,𝐹^𝑡 is admissible and𝐶^𝑡satisfies unanimity, positive respon- siveness and neutrality, then𝑃^𝑡 satisfies rank dependence.

Proof. Pick any𝑈^𝑡 such that 𝑈^𝑡

1 > 𝑈^𝑡

2 and let 𝛿 = 𝑈^𝑡

1− 𝑈^𝑡

2. Let 𝑈

′𝑡

= (𝑈^𝑡

1 − 𝛿, 𝑈^𝑡

2), then by lemma 1, 𝑃^𝐶

𝑡

1 (𝑈

′𝑡) = 𝑃^𝐶

𝑡

2 (𝑈

′𝑡) = ¹₂. Since 𝐶^𝑡 satisfies positive responsiveness and unanimity, theorem 3, together with the fact that 𝑃^𝐶

𝑡

1 (𝑈^𝑡) + 𝑃^𝐶

𝑡

2 (𝑈^𝑡) =1, implies that𝑃^𝐶

𝑡

1 (𝑈^𝑡) > 𝑃^𝐶

𝑡

1 (𝑈

′𝑡

) =𝑃^𝐶

𝑡

2 (𝑈

′𝑡

) > 𝑃^𝐶

𝑡

2 (𝑈^𝑡). □

If𝐾^𝑡 > 2 the next two propositions prove rank dependence with additional restric- tions on the team collective choice rule.

Theorem 5. If 𝐹^𝑡 is admissible and 𝐶^𝑡 is plurality rule, then 𝑃^𝐶

𝑡

satisfies rank dependence.

Proof. Consider any profile of expected payoffs𝑈^𝑡. By neutrality and admissibility we can without loss of generality label the actions such that𝑈^𝑡

1 ≥ 𝑈^𝑡

2 ≥ ... ≥ 𝑈^𝑡

𝐾^𝑡. By lemma 1, if𝑈^𝑡

𝑘 =𝑈^𝑡

𝑙, then𝑃^𝐶

𝑡

𝑘 =𝑃^𝐶

𝑡

𝑙 . Suppose𝑈^𝑡

𝑘

> 𝑈^𝑡

𝑙. The probability that any team member𝑖ranks action𝑘 highest is 𝑝_𝑘 =𝑃𝑟 𝑜 𝑏(𝑈^𝑡

𝑘+𝜖^𝑡

𝑖 𝑘−max𝑗≠𝑘{𝑈^𝑡

𝑗 +𝜖^𝑡

𝑖 𝑗} ≥0). Let𝜓 :{1, ..., 𝐾^𝑡} → {1, ..., 𝐾^𝑡}be the pairwise permutation of𝑘 and𝑙, that is the permutation that maps𝑘 to𝑙and𝑙to 𝑘and all else to itself. By exchangeability of the error terms,( (𝑈^𝑡

1+𝜖^𝑡

1, ..., 𝑈^𝑡

𝐾^𝑡+𝜖^𝑡

𝐾^𝑡) has the same joint distribution as (𝑈^𝑡

1+𝜖^𝑡

𝜓(1), ..., 𝑈^𝑡

𝐾^𝑡 +𝜖^𝑡

𝜓(𝐾^𝑡)), and so𝑈^𝑡

𝑘 +𝜖^𝑡

𝑖 𝑘 − max𝑗≠𝑘{𝑈^𝑡

𝑗 +𝜖^𝑡

𝑖 𝑗} has the same distribution as𝑈^𝑡

𝑘 +𝜖^𝑡

𝑖𝜓(𝑘) −max𝑗≠𝑘{𝑈^𝑡

𝑗 +𝜖^𝑡

𝑖𝜓(𝑗)}.

Since𝑈^𝑡

𝑘 > 𝑈^𝑡

𝑙, we have for all𝜖^𝑡

𝑖,𝑈^𝑡

𝜓(𝑘)+𝜖^𝑡

𝑖𝜓(𝑘) < 𝑈^𝑡

𝑘+𝜖^𝑡

𝑖𝜓(𝑘), and max𝑗≠𝑘{𝑈^𝑡

𝜓(𝑗)+ 𝜖^𝑡

𝑖𝜓(𝑗)} ≥ max𝑗≠𝑘{𝑈^𝑡

𝑗 + 𝜖^𝑡

𝑖𝜓(𝑗)}. By the full support assumption, it follows that 𝑃𝑟 𝑜 𝑏(𝑈^𝑡

𝑘+𝜖^𝑡

𝑖𝜓(𝑘)−max𝑗≠𝑘{𝑈^𝑡

𝑗+𝜖^𝑡

𝑖𝜓(𝑗)} ≥0) > 𝑃𝑟 𝑜 𝑏(𝑈^𝑡

𝜓(𝑘)+𝜖^𝑡

𝑖𝜓(𝑘)−max𝑗≠𝑘{𝑈^𝑡

𝑗+ 𝜖^𝑡

𝑖𝜓(𝑗)} ≥0) = 𝑝_𝑙, so𝑝_𝑘 > 𝑝_𝑙. Now, still supposing that𝑈^𝑡

𝑘

> 𝑈^𝑡

𝑙, and therefore that𝑝_𝑘 > 𝑝_𝑙, denote by(𝑛₁, ..., 𝑛_𝐾) the tuple of number of team members that rank each action first for a given ˆ𝑈^𝑡. For any choice set𝐵 ⊆ 𝐴^𝑡, let𝑉^𝐵 ={(𝑛₁, ..., 𝑛_𝐾) |∀𝑎_𝑘 ∈𝐵,∀𝑎_𝑗, 𝑛_𝑘 ≥ 𝑛_𝑗 and Í𝐾

𝑗=1𝑛_𝑗 = 𝑛} be the set of feasible ’vote’ totals that result in 𝐵 being chosen. Then, since estimated expected utilities are independent across individuals conditional on𝑈^𝑡, we can write the probability of this subset being chosen as

𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) =𝐵)=∑︁

𝑉^𝐵

𝑛!

𝑛₁!𝑛₂!...𝑛_𝐾!Π^𝐾_𝑗=1𝑝

𝑛𝑗

𝑗

Let 𝜓 : {1, ..., 𝐾^𝑡} → {1, ..., 𝐾^𝑡} be the pairwise permutation between 𝑘 and𝑙 as defined earlier. Pick any 𝐵 that contains 𝑎_𝑘 and not 𝑎_𝑙. Then (𝑛₁, ..., 𝑛_𝐾) ∈ 𝑉^𝐵

if and only if (𝑛_𝜓(1), , ..., 𝑛_𝜓(𝐾)) ∈ 𝑉^{(𝐵−{𝑎𝑘})∪{𝑎𝑙}}, the set of vote totals that results in the choice set being 𝐵, minus 𝑎_𝑘 and adding 𝑎_𝑙. Then, since 𝑛_𝑘 > 𝑛_𝑙, we have that 𝑝^𝑛𝑘

𝑘

𝑝

𝑛_𝑙 𝑙

> 𝑝

𝑛_𝑙 𝑘

𝑝^𝑛𝑘

𝑙 , so every term of the sum in 𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) = 𝐵) is greater than the corresponding term in 𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) = (𝐵− {𝑎_𝑘}) ∪ {𝑎_𝑙}). So we have for all 𝐵 containing 𝑎_𝑘 and not 𝑎_𝑙, that 𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) = 𝐵) > 𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) = (𝐵− {𝑎_𝑘}) ∪ {𝑎_𝑙}). Finally, define 𝐵₀ to be the subsets of 𝐴^𝑡 that contain neither 𝑎_𝑘 nor𝑎_𝑙, 𝐵_𝑘 the subsets containing only𝑎_𝑘, 𝐵_𝑙 the subsets containing only𝑎_𝑙 and not𝑎_𝑘, and𝐵_{𝑘 𝑙} the set containing both. Then

𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) =0× ∑︁

𝐵∈𝐵₀

𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) =𝐵) + ∑︁

𝐵∈𝐵_𝑘

1

|𝐵|𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) = 𝐵) +0× ∑︁

𝐵∈𝐵_𝑙

𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) = 𝐵) + ∑︁

𝐵∈𝐵_{𝑘 𝑙}

1

|𝐵|𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) =𝐵) 𝑃^𝐶

𝑡

𝑙 (𝑈^𝑡) =0× ∑︁

𝐵∈𝐵₀

𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) =𝐵) +0× ∑︁

𝐵∈𝐵𝑘

𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) =𝐵) + ∑︁

𝐵∈𝐵𝑙

1

|𝐵|𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) =𝐵) + ∑︁

𝐵∈𝐵𝑘 𝑙

1

|𝐵|𝑃𝑟 𝑜 𝑏(𝐶^𝑡(𝑈ˆ^𝑡) =𝐵) 𝑃^𝐶

𝑡

𝑘 and𝑃^𝐶

𝑡

𝑙 share all terms of the fourth sum, so 𝑃^𝐶

𝑡

𝑘 [𝑈^𝑡] −𝑃^𝐶

𝑡

𝑙 [𝑈^𝑡] = ∑︁

𝐵∈𝐵𝑘

1

|𝐵|𝑃𝑟 𝑜 𝑏[𝐶^𝑡(𝑈ˆ^𝑡)= 𝐵] − ∑︁

𝐵∈𝐵𝑙

1

|𝐵|𝑃𝑟 𝑜 𝑏[𝐶^𝑡(𝑈ˆ^𝑡) =𝐵] 𝑃^𝐶

𝑡

𝑘 [𝑈^𝑡] −𝑃^𝐶

𝑡

𝑙 [𝑈^𝑡] = ∑︁

𝐵∈𝐵_𝑘

1

|𝐵|[𝑃𝑟 𝑜 𝑏[𝐶^𝑡(𝑈ˆ^𝑡) =𝐵] −𝑃𝑟 𝑜 𝑏[𝐶^𝑡(𝑈ˆ^𝑡) = (𝐵− {𝑎_𝑘}) ∪ {𝑎_𝑙}]] > 0 Therefore, whenever𝑈^𝑡

𝑘

> 𝑈^𝑡

𝑙, we have𝑝_𝑘 > 𝑝_𝑙, which implies𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) > 𝑃^𝐶

𝑡

𝑙 (𝑈^𝑡).

□

Define a weighted average rule as follows:

Definition 11. A team collective choice rule 𝐶^𝑡 is a Weighted Average Rule if there exists a profile of non-negative individual voting weights, (𝑤^𝑡

1, ..., 𝑤^𝑡

𝑛^𝑡) with Í𝑛^𝑡

𝑖=1𝑤^𝑡

𝑖 = 1such that for all𝑎^𝑡

𝑘 ∈ 𝐴^𝑡 and for all𝑈ˆ^𝑡 ∈ ℜ^{𝐾𝑡𝑛𝑡}, 𝑎^𝑡

𝑘 ∈ 𝐶^𝑡(𝑈ˆ^𝑡) if and only ifÍ𝑛^𝑡

𝑖=1𝑤^𝑡

𝑖𝑈ˆ^𝑡

𝑖 𝑘 ≥ Í𝑛^𝑡 𝑖=1𝑤^𝑡

𝑖𝑈ˆ^𝑡

𝑖𝑙 for all𝑙 ≠ 𝑘 .

Theorem 6. If𝐹^𝑡is admissible and𝐶^𝑡is a weighted average rule, then𝑃^𝐶

𝑡 satisfies rank dependence.

Proof. Consider any profile of expected payoffs 𝑈^𝑡, and suppose 𝑈^𝑡

𝑘

> 𝑈^𝑡

𝑙. We have 𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) = ∫

1{Í^𝑛𝑡

𝑖=1𝑤^𝑡

𝑖𝑈ˆ^𝑡

𝑖 𝑘 ≥ max{Í^𝑛𝑡

𝑖=1𝑤_𝑖𝑈ˆ^𝑡

𝑖 𝑗}^𝐾𝑡

𝑗=1}𝑑𝐹^𝑡. Note that the prob- ability that any of these weighted averages are exactly equal is 0. Now, since

𝐹^𝑡(𝑦−𝑈^𝑡

𝑘) < 𝐹^𝑡(𝑦 −𝑈^𝑡

𝑙) for all 𝑦 ∈ ℜ, we have ˆ𝑈^𝑡

𝑖 𝑘

>_𝑠𝑡 𝑈ˆ^𝑡

𝑖𝑙, where >_𝑠𝑡 denotes thestrictfirst stochastic order, for all members𝑖. This order is closed under convo- lutions, soÍ^𝑛𝑡

𝑖=1𝑤^𝑡

𝑖𝑈ˆ^𝑡

𝑖 𝑘 >_𝑠𝑡 Í^𝑛𝑡

𝑖=1𝑤^𝑡

𝑖𝑈ˆ^𝑡

𝑖𝑙. Since1{𝑧 > 0} is increasing, non-constant and bounded, we therefore have

∫ 1{

𝑛

∑︁

𝑖=1

𝑤_𝑖𝑈ˆ^𝑡

𝑖 𝑘 ≥ max{

𝑛

∑︁

𝑖=1

𝑤_𝑖𝑈ˆ^𝑡

𝑖 𝑗}^𝐾_𝑗=1}}𝑑𝐹 >

∫ 1{

𝑛

∑︁

𝑖=1

𝑤_𝑖𝑈ˆ^𝑡

𝑖𝑙 ≥ max{

𝑛

∑︁

𝑖=1

𝑤_𝑖𝑈ˆ^𝑡

𝑖 𝑗}^𝐾_𝑗=1}}𝑑𝐹 𝑃^𝐶

𝑡

𝑘 (𝑈^𝑡) > 𝑃^𝐶

𝑡

𝑙 (𝑈^𝑡)

□