Economic design often features scenarios in which choices must be made based on stochastic inputs. In auction design, bidders drawn from a population interact with an auction mechanism, and the mechanism must then choose winners and losers of the auction. In information structure design, a principal observes information pertinent to the various actions available to one or more decision makers, and must use this information to recommend actions to the decision makers. In such settings, an important framing device is the notion of an interim rule (also called a reduced form) of an (ex-post) winner selection rule, summarizing the probability for each candidate to be selected.
We focus on the simplest and most natural class of such decision making scenarios, one which includes auctions and Bayesian persuasion  as special cases. In a winner selection environment, there is a set of candidates , each equipped with a random attribute known as its type. A winner selection rule is a randomized function (or algorithm) which maps each profile of types, one per candidate, to a choice of winning candidates, subject to the requirement that the set of winners must belong to a specified family of feasible sets. The winner selection rule is also referred to as an ex-post rule since it specifies the winning probabilities conditioned on every realized type profile. In auctions, candidates correspond to bidders, and a winner selection rule is an allocation rule of the auction. In Bayesian persuasion, candidates correspond to actions available to a decision maker, and a winner selection rule corresponds to a persuasion scheme used by a principal to recommend one of the actions to the decision maker. We restrict our attention to winner selection scenarios in which the types of different candidates are independently distributed.
We distinguish two classes of interim rules: first-order and second-order. The former is the traditional notion from auction theory, while the latter is the notion better suited for persuasion. A first-order interim rule specifies, for each candidate and type of candidate , the conditional probability of winning given that his type is . A second-order interim rule specifies more information: for each pair of candidates and type of candidate , it specifies the conditional probability of winning given that has type . First-order interim rules, when combined with a payment rule, suffice for evaluating the welfare, revenue, and incentive-compatibility of a single-item auction. For Bayesian persuasion, second-order interim rules are needed for evaluating the incentive constraints of a persuasion scheme.
Our motivation for studying winner selection at this level of generality stems from the success of Myerson’s  famous and elegant characterization of revenue-optimal single-item auctions when bidder type distributions are independent. In that special case, Myerson showed that the optimal single-item auction features a particularly structured winner-selection rule: each type is associated with a virtual value, and given a profile of reported types, the rule selects the bidder with the highest (non-negative) virtual value as the winner.
Interestingly, order sampling  works in a similar way as Myerson’s auction, in the special case when the feasible sets are sets of or fewer candidates, and each candidate has only one type. Order sampling assigns each candidate a random score variable (a die). The winners are the candidates with the highest score variables. In the language of order sampling, virtual value functions define a single-sided die for each type. Unlike Myerson’s auction, there is no notion of “revenue” or “incentive constraints” in order sampling. The task is simply to find dice that will induce a prescribed first-order interim rule.
As a generalization of order sampling and Myerson’s virtual-value approach, a dice-based winner-selection rule assigns each type a die, and selects the feasible set of winners maximizing the sum of the dice rolls. We explore the extent to which dice-based rules are applicable beyond single-parameter auctions or single-type environments, to winner selection with independent type distributions under more complex constraints. In particular, we examine whether dice-based winner-selection rules exist for winner selection subject to matroid constraints and for Bayesian persuasion.
1.1 Our Results
As mentioned previously, all of our results are restricted to settings in which the candidates’ type distributions are independent. It follows from Myerson’s characterization that every first-order interim rule corresponding to some optimal auction admits a dice-based implementation.111As we show in the appendix, there are interim rules which are not optimal for any auction. Our main result (in Section 3) is an existential proof showing that every feasible first-order interim rule with respect to a matroid constraint admits a dice-based implementation. This illustrates that the structure revealed by Myerson’s characterization is more general, and applies to other settings in which only first-order interim information is relevant. For example, single-item auctions with (public or private) budgets are such a setting (see, e.g., ). Our result also provides a generalization of order sampling from the -winner setting to the general matroid, multi-type setting.
Beyond the existential proof of dice-based implementations, we show (in Section 4) that for single-winner environments, an algorithm can construct the dice-based rule efficiently. When the types are identically distributed, we also constructively show (in Section 5) that every first-order interim rule which is symmetric across candidates admits a symmetric dice implementation; i.e., different candidates have the same die for the same type. This is consistent with Myerson’s symmetric characterization of optimal single-item auctions with i.i.d. bidders, and generalizes it to any other first-order single-winner selection setting in which candidates are identical. Single-item auctions with identically distributed budgeted bidders are such a setting, and a symmetric dice-based implementation of the optimal allocation rule was already known from .
For single-winner cases, we also show the converse direction: how to efficiently compute the first-order interim rule of a given dice-based winner selection rule. In effect, these results show that collections of dice are a computationally equivalent description of single-winner first-order interim information. This implies a kind of equivalence between the two dominant approaches for mechanism design: the Myersonian approach based on virtual values (i.e., dice), and the Borderian approach based on optimization over interim rules.
In an attempt to leverage the same kinds of insights for Bayesian persuasion, we examine (in Sections 6 and 7) the dice implementability of second-order interim rules. When the candidate type distributions are non-identical, we show an impossibility result. We construct an instance of Bayesian persuasion with independently distributed non-identical actions, and show that no optimal persuasion scheme for this instance can be implemented by dice. Since second-order interim rules are sufficient for evaluating the objective and constraints of Bayesian persuasion, this implies that there exist second-order interim rules which are not dice-implementable. This rules out the Myersonian approach for characterizing and computing optimal schemes for Bayesian persuasion with independent non-identical actions, complementing the negative result of  which rules out the Borderian approach for the same problem.
Our impossibility result disappears when the actions are i.i.d., since second-order interim rules collapse to first-order interim rules in symmetric settings. In particular, as we show in Section 7, our results for first-order interim rules, combined with those of , imply that Bayesian persuasion with i.i.d. actions admits an optimal dice-based scheme, which can be computed efficiently.
1.2 Additional Discussion of Related Work
Myerson  was the first to characterize revenue-optimal single-item auctions; this characterization extends to single-parameter mechanism design settings more generally (see, e.g., ). The (first-order) interim rule of an auction, also known as its reduced form, was first studied by Maskin and Riley  and Matthews . The inequalities characterizing the space of feasible interim rules were described by Border [3, 4]
. Border’s analytically tractable characterization of feasible interim rules has served as a fruitful framework for mechanism design, since an optimal auction can be viewed as the solution of a linear program over the space of interim rules. Moreover, this characterization has enabled the design of efficient algorithms for recognizing interim rules and optimizing over them, byCai et al.  and Alaei et al. . This line of work has served as a foundation for much of the recent literature on Bayesian algorithmic mechanism design in multi-parameter settings.
It is important to contrast our dice-based rule with the characterization of Cai et al. . In particular, the results of Cai et al.  imply that every first-order interim rule can be efficiently implemented as a distribution over virtual value maximizers. In our language, this implies the existence of an efficiently computable dice-based implementation in which the dice may be arbitrarily correlated. Our result, in contrast, efficiently computes a family of independent dice implementing any given first-order interim rule in single-winner settings, and shows the existence of a dice-based rule in matroid settings. This is consistent with Myerson’s characterization, in which virtual values are drawn independently.
Alaei et al.  also studied winner-selection environments, under the different name “service based environments.” For single-winner settings, they proposed a mechanism called stochastic sequential allocation (SSA). The mechanism also implements any feasible first-order interim rule, by creating a token of winning and transferring the token sequentially from one candidate to another, with probabilities defined by an efficiently computed transition table. Dice can be considered as the special case of SSA in which the transition probabilities are independent of the current owner of the token.
As another motivation for our focus on dice-based rules, order sampling studies how to sample winners from candidates with given inclusion probabilities (i.e., implement an interim rule), by assigning a random score variable (die) to each candidate. Rosén  showed that parameterized Pareto distributions can be used to implement a given interim rule asymptotically. Aires et al.  proved the existence of an order sampling scheme that exactly implements any feasible interim rule. Our existential proof is a generalization of the proof of  to settings with multiple types and matroid constraints.
The Bayesian persuasion model is due to Kamenica and Gentzkow , and is the most influential model in the general space of information structure design (see the survey by Dughmi  for references). Bayesian persuasion was examined algorithmically by Dughmi and Xu , who observed its connection to auction theory and interim rules, and examined the computational complexity of optimal schemes through the lens of optimization over interim rules.
Of particular relevance to our work is the negative result of Dughmi and Xu  for Bayesian persuasion with independent non-identical actions: it is #P-hard to compute the interim rule (first- or second-order) of the optimal scheme, or more simply even the sender’s optimal utility. Most notable about this result is what it does not rule out: an algorithm implementing the optimal persuasion scheme “on the fly,” in the sense that it efficiently samples the optimal scheme’s (randomized) recommendation when given as input the profile of action types. Stated differently, the negative result of Dughmi and Xu  merely rules out the Borderian approach for this problem, leaving other approaches — such as the Myersonian one — viable as a means of obtaining an efficient “on the fly” implementation. This would not be unprecedented: Gopalan et al.  exhibit a simple single-parameter auction setting for which the optimal interim rule is #P hard to compute, yet Myerson’s virtual values can be sampled efficiently and used to efficiently implement the optimal auction. Our negative result in Section 7 rules out such good fortune for Bayesian persuasion with independent non-identical actions: there does not exist a (Myersonian) dice-based implementation of the optimal persuasion scheme in general.
2.1 Winner Selection
Consider choosing a set of winners from among candidates. Each candidate has a type , drawn independently from a distribution . A winner-selection rule maps each type profile , possibly randomly, to one of a prescribed family of feasible sets . When , we refer to as a winning candidate, and to as his winning type. Writing for the (independent) joint type distribution, we also refer to as the winner-selection environment. When is the family of singletons, as in the setting of the single item auction, we call a single-winner environment.
This general setup captures the allocation rules of general auctions with independent unit-demand buyers, albeit without specifying payment rules or imposing incentive constraints. Moreover, it captures Bayesian persuasion with independent action payoffs, albeit without enforcing persuasiveness (also called obedience) constraints.
In this paper, we focus on settings in which the feasible sets are the independent sets of a matroid. We use the standard definition of a matroid as a pair , where is the ground set and is a family of so-called independent sets, satisfying the three matroid axioms. We also use the standard definitions of a circuit and rank function . The restriction of to some is the matroid .222Note that we deviate slightly from the standard definition in that we do not restrict the ground set. For details on matroids, we refer the reader to Oxley .
A matroid is separable if it is a direct sum of two matroids and . Namely, , . Note that if is non-separable, then ; otherwise is the direct sum of singleton matroids. We use the following theorem.
[Whitney ] (1) When is a non-separable matroid, for every , there is a circuit containing both and . (2) Any separable matroid is a direct sum of two or more non-separable matroids called the components of .
In most of the remainder of the paper, we focus on winner selection environments where is the family of independent sets of a matroid . We therefore also use to denote the environment.
2.3 Interim Rules and Border’s Theorem
A (first-order) interim rule specifies the winning probability for all in an environment . More precisely, we say that a winner-selection rule implements the interim rule for a prior if it satisfies the following: if the type profile is drawn from the prior distribution , then . An interim rule is feasible (or implementable) within an environment if there is a winner-selection rule implementing it that always outputs an independent set of .
2.3.1 Border’s theorem and implications for single-winner environments
The following theorem characterizes the space of feasible interim rules for single-winner settings.
The following result leverages Theorem 2.3.1 to show that efficient algorithms exist for checking the feasibility of an interim rule, and for implementing a feasible interim rule.
[[5, 2]] Given explicitly represented priors and an interim rule in a single-winner setting, the feasibility of can be checked in time polynomial in the number of candidates and types. Moreover, given a feasible interim rule , an algorithm can find a winner-selection rule implementing in time polynomial in the number of candidates and types.
In our efficient construction for single-winner settings, we utilize a structural result which shows that checking only a subset of Border’s constraints suffices [3, 16, 5]. This subset of constraints can be identified efficiently.
[Theorem 4 of ] An interim rule is feasible for a single-winner setting if and only if for all possible , the sets satisfy the following Border’s constraint:
When the candidates’ type distributions are i.i.d., i.e., and are the same for all candidates , it is typically sufficient to restrict attention to symmetric interim rules. For such rules, is equal for all candidates . In the i.i.d. setting, we therefore notationally omit the dependence on the candidate and let refer to the common type set of all candidates, to the candidates’ (common) type distribution, and to the probability that a particular candidate wins conditioned on having type . In the i.i.d. setting, only the symmetric constraints from Theorem 2.3.1 suffice to characterize feasibility ; namely,
2.3.2 Border’s Theorem for matroid environments
For general settings with matroid constraints, Alaei et al.  established the following generalized “Border’s Theorem.”
[Theorem 7 of ] Let map each type to the (unique) candidate with . An interim rule is feasible within an environment if and only if for all possible type subsets ,
where . In later sections, we omit the function , and for any type set just write instead of .
2.4 Winner-Selecting Dice
We study winner-selection rules based on dice, as a generalization of order sampling to multiple types and general constraints. A dice-based rule fixes, for each type , a distribution over real numbers, which we call a die. Given as input the type profile , the rule independently draws a score for each candidate by “rolling his die;” it then selects the feasible set of candidates maximizing the sum of scores as the winner set, breaking ties with a predefined rule. In this paper, we will mainly discuss matroid feasibility constraints, for which a feasible set maximizing the sum of scores can be found by a simple greedy algorithm: candidates are added to the winner set in decreasing order of their scores, breaking ties uniformly at random, as long as the new winner set is still an independent set of the matroid and their scores are positive. When candidates have the same type sets, we call a dice-based rule symmetric if is the same for all .
Myerson’s optimal auction is a dice-based winner-selection rule. In Myerson’s nomenclature, is candidate ’s virtual value, and is a single-sided die with the virtual value.
Let be the set of all types of all candidates and
be a vector of dice, one per type. Given an interim rule, and a winner-selection environment , we say that implements , or describes winner-selecting dice for in , if the dice-based rule given by implements within the environment .
2.5 Second-order Interim Rules
A (first-order) interim rule, as defined in Section 2.3, specifies, for each candidate , the conditional type distribution of in the event that is chosen as the winner. We define a second-order interim rule333Our notion of second-order interim rules is different from the notion defined in . Because Cai et al.  consider correlation in types, their notion of second-order interim rules is aimed at capturing the allocation dependencies arising through such type correlation, rather than solely through the mechanism’s choice. which maintains strictly more information, as needed for describing the incentive constraints of Bayesian persuasion. Such a rule specifies, for each pair of candidates and (where may or may not be equal to ), the conditional type distribution of in the event that is chosen as the winner. Formally, a second-order interim rule specifies for each pair of candidates , and type . We say that a winner-selection rule implements for a prior if it satisfies the following: if the type profile is drawn from the prior distribution , then . A second-order interim rule is feasible if there is a winner selection rule implementing it.
3 Existence of Dice Implementation for Matroids
In this section, we prove our first theorem:
Let be a matroid winner selection-environment with a total of types, and let be an interim rule that is feasible within . There exist winner-selecting dice , each of which has at most faces, which implement .
The proof consists of two parts. First, we generalize the result of , which showed the existence of continuous winner-selecting dice for feasible interim rules for a -uniform matroid with fixed types, to general matroids and multiple types. Second, we convert the continuous dice to dice with at most faces each, while keeping the interim probabilities unchanged.
3.1 Continuous Winner-Selecting Dice
Let be a matroid, and a feasible interim rule within the winner-selection environment . There exist winner-selecting dice over that implement in .
We assume without loss of generality that the candidates’ type sets are disjoint, and use to denote the set of all types. We use and as shorthand for and , where is the candidate for whom . Moreover, given a set of types , we write . Recall the Border constraints
where is the expected rank of types in which show up, a submodular function over the type set . The Border constraints can therefore be interpreted as follows: An interim rule is feasible for and if and only if is in the polymatroid given by , where . Equivalently, is feasible if and only if the submodular slack function is non-negative everywhere.
When is feasible, we call a set tight for if the Border constraint (3) corresponding to is tight at , i.e., . By definition, is always tight. The family of tight sets, being the family of minimizers of the submodular slack function, forms a lattice: the intersection and the union of two tight sets is a tight set.
The tightness of a set means that the expected number of winners from equals the expected rank of types in which show up. In other words, is tight if and only if a maximum independent subset of is always selected as winners.
By Remark 3.1, the types in minimal non-empty tight sets need to be treated preferentially, i.e., assigned higher faces on their dice, compared to types outside them. Because they play such an important role, we define them as barrier sets. Formally, we define the set of active types to be the types who win with positive probability. Barrier sets are subsets of . If there is at least one non-empty tight set of active types, we define the barrier sets as the (inclusion-wise) minimal such sets. Otherwise, we designate the entire set of active types as the unique barrier set.
To prove Theorem 3.1, we first assume that the matroid is non-separable. Separable matroids will be handled in the proof of Theorem 3.1 by combining the dice of their non-separable components. Because barrier sets get precedence, we first show how to construct dice for barrier sets with Lemma 3.1. Once we have dice for barrier sets, we can repeatedly “peel off” the tight sets and combine their dice, which is captured in Lemma 3.1. We start with the existence of dice for barrier sets:
Let be a non-separable matroid, and a feasible interim rule within the winner-selection environment . Let be a barrier set for , and define to be . There exists a vector of distributions over , such that implements in .
Without loss of generality, assume that the barrier set contains types, and number them by .
When and is tight, all types in belong to the same candidate, so there is no competition between candidates. Because of the tightness of , we can simply assign a single-sided die with face value to each type in . Thus in the proof, we assume that either there is more than one candidate with a type in or that is not tight.444Recall that could be the (possibly non-tight) barrier set of all active types, if no non-empty set is tight. In both cases, this implies that for all . For if , the singleton would be a proper tight subset of , which contradicts the assumption that is a barrier set.
We create from any continuous distribution555 For example, one can use an exponential distribution.
For example, one can use an exponential distribution.with support . For each type , we assign a parameter . Also, we choose a global parameter . To sample from , we draw a primitive roll and output .
We define to be the interim probability of type winning in the environment when the winner-selecting dice with parameters and are used for . Each is continuous in all of its parameters. Using for short, consider the following system of equations with variables :
The objective of the proof is to show that the system of equations (5) admits a solution. This solution will directly induce a dice system that implements for types in . Notice that the system is redundant: Equation (4) is implied by the Equations (5). Throughout the proof, we use to denote the prefix and to denote the suffix of parameters . In particular, . We prove the following inductively for :
Applying the claim with , for every positive , there is a solution that satisfies the first equations. Because of the redundancy of the system, satisfying the first equations guarantees that the last equation is also satisfied.
For the base case , consider an arbitrary positive . We need to prove the existence of a such that . When , all types get non-negative die rolls. Therefore, a maximum independent subset of is always selected. Thus, is tight at , i.e., . On the other hand, as increases, the probability that all die rolls are negative goes to 1, meaning that in the limit, no agent wins. Thus, . Furthermore, each strictly and continuously decreases with . Because , by the Intermediate Value Theorem, for every , there is a unique such that . We denote this unique by ; notice that is a continuous function of . When is a tight barrier set, i.e., , the equation is satisfied for , so . This establishes the base case of the induction hypothesis.
For the inductive step, fix an arbitrary , and let be arbitrary. For any fixed , by induction hypothesis, there is a unique such that satisfies the first equations of (4)–(5). Lemma 3.1 below shows that there is a unique such that satisfies the first equations of (4)-(5). The inductive claim now follows by defining ∎
Fix any feasible . For every and suffix , there is a unique such that .
The proof of Lemma 3.1 is based on large part on the following monotonicity properties:
Assume that contains types of more than one candidate or is not tight. For any , consider as a function of , for . The function satisfies the following properties:
It is weakly decreasing in , for all .
It is strictly increasing in .
If , then is strictly decreasing in . If in addition and , then is also strictly decreasing in .
For notational convenience, we define the shorthand . Furthermore, for fixed (which will be clear from the context), we define to be the t component of , for any .
Lemma 3.1 As we did in the proof of Lemma 3.1 for the base case of the induction claim, we will examine the limits of as and . By doing so, we will establish that lies between the two limits. Then, using the continuity and strict monotonicity (by Lemma 3.1) of , the Intermediate Value Theorem implies that there is a unique satisfying .
To compute , consider the set of types . As , with probability approaching , will dominate all other types. We distinguish two cases, based on whether is finite or not.
If , the parameter will be finite in the limit. Therefore, a maximum independent subset of will be selected as winners, implying that is tight in the limit. Because ensures that the first equations of (4)–(5) are satisfied, each type wins with probability . Because is tight, . From the Border constraint for the set , ; otherwise would be a tight set, contradicting that is a barrier set. Rearranging, we have shown that .
If , then all types will get negative die rolls with probability approaching 1, so for all types . Thus, , i.e., , because all types in a barrier set are active by definition.
To compute , define the type set . As , with probability approaching , will dominate all other types because for , goes to . Again, we distinguish two cases, based on whether or not.
If , then in the limit, a maximum independent subset of must always be chosen as winners, so approaches tightness. Again, ensures that for , so we have . Combined with , which is also ensured by , we obtain that
Rearranging the Border constraint corresponding to gives us that because is not tight. Adding on both sides, we get . Finally, canceling out , we obtain that .
If , as , type will get a negative roll with probability approaching , so .
In summary, we have shown that . Because is continuous and monotone in , by the Intermediate Value Theorem, there exists a unique such that .
In the proofs for Lemmas 3.1 and 3.1, we will often want to analyze the effect of keeping for all unchanged, while changing to . Thereto, we always consider the following coupling of dice rolls in the two scenarios, which we call primitive-roll coupling for : all dice will obtain the same primitive rolls under the two scenarios, but scale them differently. More precisely, we consider the rolls and , for any given primitive rolls , where for . Recall that all for are functions of .
Part 1 of Lemma 3.1 To show that is weakly decreasing in for all , we again use induction on . The induction hypothesis is the following:
Each entry of is weakly increasing in for .
is weakly decreasing in for .
We begin with the base case . The first part of the base case — that is weakly increasing in — has been shown in the proof of Lemma 3.1. To prove the second part of the base case, we use primitive-roll coupling for with . For every scenario with primitive rolls in which is not a winner with , we show that cannot become a winner with .
Increasing to can only (weakly) increase the threshold , by the first part of the base case.
Let be the set of types with rolls higher than the roll of type when the rolls are derived from using . Since is not winning, we have
The inequality is due to the submodularity of the matroid rank function, and shows formally that (potentially) adding to the set of types with higher rolls than cannot help become a winner.
For the induction step, consider some and fix a . By the induction hypothesis, each entry of is weakly increasing in for , and is weakly decreasing in for .
We first show that each entry of is weakly increasing in for . The key is component of . By the second part of the induction hypothesis, applied with , is weakly decreasing in . Since is defined as the winning probability of with the given parameters, when is raised to , in order to keep , we require . By applying the induction hypothesis twice, once for and once for , all entries of weakly increase. Therefore, all entries of are weakly increasing in all for .
To prove the second part of the inductive step, recall that . For , is a constant, and in particular weakly decreasing in . Consider some . First, is weakly increasing in by the first part of the induction hypothesis. By the second part of the induction hypothesis, is weakly decreasing in all of its variables except , so substituting shows that is weakly decreasing in .
Part 2 of Lemma 3.1 Recall that . To show that is strictly increasing in , we will show that at least one type has strictly decreasing in . By Part 1 of the lemma, each for is weakly decreasing in . By definition, ensures that the first equations of the system (4)-(5) are satisfied; this implies that . Thus, if at least one of the is strictly decreasing in , to keep the summation unchanged, must increase strictly in . We consider two possible cases for , as permitted by the assumption of the lemma:
is not tight. In this case, we first show that is a strictly increasing function of . Consider the primitive-roll coupling for . For any primitive rolls , the number of winners will not decrease when increases to . Thus, we can bound the summation . Because is not tight, with non-zero probability, is such that for all . And because is drawn from a continuous distribution, with positive probability, as well. In that case, is the only candidate with a positive die roll. This implies that . Because is defined as the unique satisfying the equation , we obtain that .
Now consider any primitive rolls under which a type wins, and is such that but . Such rolls must occur with positive probability, because all the dice are fully supported over . In that case, is no longer a winning type under . Thus is strictly decreasing in .
There is more than one candidate, i.e., there is a type with .
If there is such a type with , then is strictly decreasing in by the second part of Lemma 3.1.
Otherwise, all types have . Define for all entries except type , where it equals . Consider a type with . By definition of , we get that . And by the first part of Lemma 3.1, . So wins with strictly higher probability under the parameters than under the parameters . As shown in Part 1 of Lemma 3.1, is weakly decreasing in all of its variable except . Thus, the only way that the winning probability of agent can increase is for the i component of to be strictly greater than the i component of .
Finally, consider some type with . Because all components of weakly increase going from to , and the i component strictly increases, we get that . Thus, we have shown that there is a such that is strictly decreasing in .
Lemma 3.1 We prove both statements using primitive-roll coupling. For the second part of the lemma, notice that when is increased to for , all components of weakly increase. Thus, all components of are weakly larger than those of , and , while . Thus, we can apply primitive-roll coupling in both cases.
Since the matroid is non-separable, there is a circuit that contains and . Under the parameter vector , with non-zero probability (over primitive rolls ), the candidates in get the highest rolls, (barely) wins with the second-lowest roll, and gets the lowest roll among all candidates in and does not win. When increases to , with non-zero probability, the scaled roll for type becomes the lowest so that ceases to be a winner. Combining this with Part 1 of Lemma 3.1 (which states that is weakly decreasing in ), we have that is strictly decreasing in .
To generalize Lemma 3.1 to arbitrary sets of types, we will need a construction that allows us to “scale” the faces of some dice such that they will always be above/below the faces of another set of newly introduced dice; such a construction will allow us to give dice for types in barrier sets higher faces than other dice. For the types of full-support distributions over we have been using so far, this would be impossible. There is a simple mapping that guarantees our desired properties: we map faces from to the set by mapping all positive , and mapping all negative to . Notice that the new dice implement the same interim rule as the old ones: in matroid environments, the set maximizing the sum of die rolls is determined by the greedy algorithm, and hence, only the relative order between die faces matters. With the help of this mapping, we prove the following lemma, similar to Lemma 3.1.
Let be a feasible interim rule within a winner-selection environment , where is a non-separable matroid. Fix a tight set , and let be a minimal tight set that includes as a proper subset, if such a set exists; otherwise let