# Characterization of Incentive Compatibility of an Ex-Ante Constrained Player

We consider a variant of the standard Bayesian mechanism, where players evaluate their outcomes and constraints in an ex-ante manner. Such a model captures a major form of modern online advertising where an advertiser is concerned with her/his expected utility over a time period and her/his type may change over time. We are interested in the incentive compatibility (IC) problem of such Bayesian mechanism. Under very mild conditions on the mechanism environments, we give a full characterization of IC via the taxation principle and show, perhaps surprisingly, that such IC mechanisms are fully characterized by the so-called auto-bidding mechanisms, which are pervasively fielded in the online advertising industry.

## Authors

• 1 publication
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## 1 Introduction

In the probability space consisting of all auction events over the given time period, the constraints imposed by such contracts are expressed in an ex-ante manner. That is, satisfaction of the constraint is evaluated by taking the expectation over the whole probability space. Most constraints in classic mechanism design literature, however, are

interim constraints, which are separately evaluated for each possible type of each player. When an ex-ante constraint on overall expectation is binding, it is possible for the player to design more complicated misreport strategy: instead of separately maximizing the utility for each of her own type, the player can make contingent plans about the type report over her whole type space, so that her constraint remains satisfied, but her utility is overall better off than always reporting truthfully. Such possibility poses new challenges to the design of incentive compatible mechanisms.

In this paper, we analyze the general mechanism design problem with an ex-ante constraint on the outcome and payment, which is not limited to the design of single-item auctions. From a fixed player’s perspective, the mechanism is abstracted into a pair of interim rules (but with ex-ante utility model and constraints), and we characterize interim rules of any incentive compatible mechanism. Our first characterization in Section 4 states that, it is without loss of generality to use the following implementation of incentive compatible mechanism: the designer maximizes some new value-minus-payment utility on behalf of the player for every type realization of the mechanism, where the new value is defined by linearly combining a utility term and a constraint term using some factors that are fixed across all type realization of the mechanism. For example, when the player is a value maximizer with budget constraint, it suffices to determine factors and maximize times value minus times payment. This characterization justifies the auto-bidding technique in online advertising, where an advertiser’s evaluations for the advertisement slot’s value and contribution to constraint are combined linearly to give input to an auction. Our second characterization, given in Section 6, shows that incentive compatibility can be characterized using a multi-dimensional payment identity, and the new payment identity differs from the one in classic quasi-linear mechanism design literature Myerson (1981); Manelli and Vincent (2007) only by linearly combining the utility and constraint terms.

## 2 Related works

Many recent works study the the mechanism design problems for players with various ex-ante constraints, typically under the context of online advertisement auction. Unlike the widely used constraint of individual rationality which is separately evaluated for every possible type of the player (interim IR) Ashlagi et al. (2016) or every profile of all players’ types (ex-post IR) Ashlagi et al. (2019), an ex-ante constraint is evaluated by taking expectation over the whole game’s probability space. Two of the most widely discussed ex-ante constraints are budget constraint Chen et al. (2021); Zhou et al. (2008), which is to limit the overall expectation of payment (the term ‘budget constraint’ is also used for interim or ex-post constraint on payment Burkett (2016), and the budget level can be private Che and Gale (1998), but in this paper we focus on an ex-ante constraint known to the designer), and ROI constraint, which is to limit the ratio given by the player’s expected utility over her expected payment Balseiro et al. (2021); Golrezaei et al. (2021b); Heymann (2019). Given the constraint, the player seeks to maximize her utility without violating the constraint. In many works the player’s utility is still given by the standard quasi-linear utility, which is the value she wins from the outcome minus payment. Another player model which becomes increasingly popular assumes that the player’s utility is just the value she wins, and the trade-off between value and payment is completely controlled by the ex-ante constraint Babaioff et al. (2021). This model plus a budget constraint resembles the classic model of Fisher’s market Brainard and Scarf (2005), where the player maximizes the value of items she wins, under the constraint that the money spent on the items does not exceed a given limit.

To solve the mechanism design problem with ex-ante constrained players, the technique called auto-bidding is widely proposed Balseiro et al. (2021); Aggarwal et al. (2019); Deng et al. (2021); Wen et al. (2021); Li et al. (2020). Typically, the mechanism is implemented by running an auction which repeatedly asks for a single bid from the player, and decide allocation and payment according to the bid. Auto-bidding repeatedly decides the bid on behalf of the player, so that the total utility over multiple rounds is maximized without violating the constraint. An important result is that, under certain settings, the optimal bidding rule is to always linearly combine the current item’s contribution to utility and contribution to the constraint using some factors that are fixed across all rounds. Many previous works obtain this result under different restrictions: Aggarwal et al. (2019) assumes finite rounds of game and a posted price auction, and Balseiro et al. (2021); Babaioff et al. (2021) assumes specific forms of the player’s utility and the ex-ante constraint. Some other works capture the sequential revelation of players’ types, and design dynamic mechanisms or control strategies that achieve low regret over multiple rounds Golrezaei et al. (2021a); Balseiro and Gur (2019).

Another literature investigates the technique called pacing. Pacing can be similarly seen as certain control technique deciding input to a repeated auction, but its goal is often to smooth the usage of the player’s budget over a certain time period Agarwal et al. (2014). Smoothing the budget usage does not necessarily involve the maximization of the player’s utility, which makes it less natural to define the corresponding incentive compatibility problem. However, some works define pacing as trying to use up the budget by uniformly scaling the bid Conitzer et al. (2018); Balseiro et al. (2017), which actually maximizes the player’s utility given some certain player models.

## 3 Model

To study the incentive compatibility of the mechanism, we focus our attention to an arbitrarily chosen player and analyze the incentive problems from her own perspective, without concerning any component of the mechanism that does not affect the chosen player’s action. Methodologically, all analysis in the paper considers the mechanism’s interim rules with respect to a single fixed player. When there are multiple players, all the results in this paper can be applied to each of them, so that each player has her corresponding incentive compatibility characterization about her interim rules.

Denote by the player’s type space, by the mechanism’s outcome space. When analyzing incentive compatibility, it is without loss of generality to consider only direct mechanisms, which take the player’s type report as the only input from the player. Therefore, from the player’s perspective, we consider a pair of interim rules as the mechanism’s abstraction, where denotes the mapping from type report to outcome, and denotes the mapping from type report to expected payment. The outcome space are chosen so that both and are deterministic functions: any possible lottery over outcomes must be encoded by a corresponding single point in . For example, if we choose , any possible lottery to allocate two items to the player can be encoded by a single point in .

Denote by the set of Borel probability measures on . The player’s type distribution is given by some probability measure . We say that a subset has positive measure if , and a condition holds for almost every if every subset where the condition fails does not have positive measure. Unless otherwise specified, all expectations in this paper are taken with respect to .

Denote by a strategy the player can take. The player model is given by the combination of a utility function and an ex-ante constraint, and is encoded by tuple . We assume that the player is risk-neutral, so that the utility is also evaluated by taking ex-ante expectation. The utility extracted from the outcome is given by function , and overall utility is given by the ex-ante expectation of utility from outcome minus a constant times payment. Given interim rules and report strategy , the ex-ante utility is given by

 U(x,p,s)\coloneqqEv,v′∼s(v)[u(x(v′),v)−c1⋅p(v′)]. (1)

Note that for any , we can rescale into , so that maximizing is equivalent to maximizing . Therefore, it is without loss of generality to consider instead of .

The ex-ante constraint is decided by function , constant and constant . The constraint is given by:

 Ev,v′∼s(v)[f(x(v′),v)−c2⋅p(v′)]≥C. (2)

And we say that the ex-ante constraint is binding when reporting truthfully, if

 Ev[f(x(v),v)−c2⋅p(v)]=C

Throughout this paper, we only analyze strategy for which the expectation in equation (1) and (2) exist. In fact, it is sufficient to consider strategies which are some random lotteries over a finite number of deterministic report rules , with each being measurable.

An ex-ante constraint can be thought of as a contract between the mechanism designer and the player, so that even without any prior distributional knowledge, the player is convinced that the overall outcome and payment are under some specific control. Our definition of incentive compatibility requires that the constraint is always satisfied by truth-telling, therefore an alternative interpretation is that when violating the constraint, the player’s utility is . This player model covers a wide range of buyer models in online advertising and auto-bidding literature, including:

• Value maximizer with ROI constraint: letting , where is the lowest acceptable ROI for the buyer.

• Utility maximizer with budget constraint: letting , , and be the buyer’s budget limit.

Also, the model naturally covers the buyer of a multi-item auction. With equal to the number of items and , we can use each dimension of to represent the player’s valuation for a subset of the items, and each dimension of to represent the allocation probability of an item.

We make the following technical assumptions on , , functions of the player model , and the interim rules .

###### Assumption 1.

is a compact subset of , and is a compact subset of . are all continuous.

By assuming compact and continuous , we make the values of bounded.

We focus on deriving characterizations of the mechanism’s incentive compatibility. The incentive compatibility for the player with ex-ante constraint is defined as follows:

###### Definition 1.

An interim rule pair is incentive compatible, if

 Ev[f(x(v),v)−c2⋅p(v)]≥C,

AND, for any strategy s.t. the expectations and exist, at least one of the following two inequalities hold:

1. .

2. .

As explained in the beginning of the section, we abstract the mechanism into a pair of interim rules . For incentive compatible interim rule pair , reporting truthfully does not violate the ex-ante constraint, and maximizes utility among all report strategies that does not violate the constraint.

## 4 Auto-bidding and taxation principle

Following the idea of taxation principle, we give our first characterization of incentive compatibility. Denote by the range of . The classic taxation principle tells that, incentive compatibility mechanisms can be implemented by designing payment rule that is some mapping from to , and decide outcome by choosing the optimal outcome in on behalf of the player. Theorem 1 tells that, a similar implementation can be used for player with an ex-ante constraint:

###### Theorem 1.

Under Assumption 1, interim rule pair is incentive compatible if and only if at least one of the following two conditions holds:

1. The ex-ante constraint is binding when reporting truthfully, and there exists function s.t. for almost every ,

 px(x(v))=c2⋅p(v),

and for almost every ,

 x(v)∈argmaxq∈Qx[f(q,v)−px(q)], (3)

and if there exists s.t. , then we have .

2. There exists constant and function s.t. for almost every ,

 px(x(v))=(c1+rc2)p(v),

and for almost every ,

 x(v)∈argmaxq∈Qx[u(q,v)+r⋅f(q,v)−px(q)], (4)

and if , the ex-ante constraint is binding when reporting truthfully.

Theorem 1 can be seen as a generalization of the widely used auto-bidding technique: the game runs by offering a price for every possible outcome , and to report optimally on behalf of the player, the controller either maximizes contribution to constraint per round , or maximizes a goal which is some fixed linear combination of the utility and the contribution to constraint. We show in Theorem 3 that, under some additional assumptions, the corresponding interim rules are characterized by an equality similar to the multi-dimensional version of standard payment identity, which generalizes the discovery in auto-bidding literature, that the underlying auction mechanism should be truthful for quasi-linear players.

To prove Theorem 1, we first provide some useful lemmas. For any , define set and by

 V+(r)\coloneqq {v|v∈V,∃v′∈V: f(x(v′),v)−c2⋅p(v′)−r⋅[f(x(v′),v)−c2⋅p(v′)] +r⋅[f(x(v),v)−c2⋅p(v)]} V−(r)\coloneqq {v|v∈V,∃v′∈V: f(x(v′),v)−c2⋅p(v′)>f(x(v),v)−c2⋅p(v), [u(x(v′),v)−c1⋅p(v′)]−[u(x(v),v)−c1⋅p(v)] >−r⋅[f(x(v′),v)−c2⋅p(v′)] +r⋅[f(x(v),v)−c2⋅p(v)]}

The following lemma tells that, such subsets of are measurable.

###### Lemma 1.

Under Assumption 1, for any , and are both measurable.

###### Proof.

For any , we prove that is open subset of , and similarly is open subset of , therefore both sets are measurable.

By continuity of , for any s.t. and , there exists some so that for any where is the open ball centered at with radius , we have and . This gives , therefore is open subset of . ∎

And the following lemma allows us to construct some measurable deterministic deviation:

###### Lemma 2.

Under Assumption 1, for any with , for any , any and , suppose for every , there exists s.t. for any , we have that

 γi(v,v′)\coloneqq αi⋅[u(x(v′),v)−c1⋅p(v′)] − αi⋅[u(x(v),v)−c1⋅p(v)] + βi⋅[f(x(v′),v)−c2⋅p(v′)] − βi⋅[f(x(v),v)−c2⋅p(v)]

is positive, then there exists function s.t. for any and function , the integration exists and is positive.

###### Sketched proof.

gives that, we can find some and some with , so that for any , there exists s.t. any , we have . By compactness of , we can obtain that is open in , therefore is closed in .

For any and any , since are all continuous, for every we can find such an open set that contains : there exists some s.t. for every and every , we have . The set

 {Ev|v∈Eϵ}

is an open cover of . Since is compact and is closed in , we can find some finite subcover of . By assigning each to some element of the finite subcover, we can define function whose range is finite. For we can simply make . This makes the integration of exist for every . ∎

Given the above two lemmas, we are able to prove the following lemma, which is essential for our characterizations of incentive compatibility. In fact, the lemma itself gives if and only if condition of incentive compatibility.

###### Lemma 3.

Under Assumption 1, the interim rule pair is incentive compatible, if and only if , and at least one of the following two conditions holds:

1. The ex-ante constraint is binding when reporting truthfully, and for almost every , for every , , and if , we have .

2. There exists s.t. for almost every , for every ,

 [u(x(v′),v)−c1⋅p(v′)]−[u(x(v),v)−c1⋅p(v)] (5) ≤ −r⋅[f(x(v′),v)−c2⋅p(v′)] +r⋅[f(x(v),v)−c2⋅p(v)],

and if , the ex-ante constraint is binding when reporting truthfully.

###### Proof.

Sufficiency: Suppose the first condition holds. A strategy obtains strictly higher utility than reporting truthfully only if there exists some with s.t. for every , it reports some with positive probability, and . However, for almost every and any such we have , which means such a strategy violates the ex-ante constraint. Therefore, no strategy gets higher utility than truthfully reporting without violating the constraint.

Suppose the second condition holds. If , reporting truthfully maximizes ex-post utility for any . Therefore, every does not obtain higher expected utility than truthfully reporting.

If , suppose there exists strategy s.t. reporting according to does not violate the ex-ante constraint, but gives strictly higher utility than reporting truthfully. This means, for any , we have

 Ev,v′∼s(v)[u(x(v′),v)−c1⋅p(v′)] > Ev[u(x(v),v)−c1⋅p(v)], r⋅Ev,v′∼s(v)[f(x(v′),v)−c2⋅p(v′)] ≤ r⋅Ev[f(x(v),v)−c2⋅p(v)]

By subtraction between the equations and rearrangement of terms, we have:

 Ev,v′∼s(v)[u(x(v′),v)−c1⋅p(v′)] −Ev[u(x(v),v)−c1⋅p(v)] > −r⋅Ev,v′∼s(v)[f(x(v′),v)−c2⋅p(v′)] +r⋅Ev[f(x(v),v)−c2⋅p(v)]

However, we have equation (5) for almost every and every , which leads to contradiction. We conclude that such does not exist, which means is incentive compatible.

Necessity: Suppose is incentive compatible, but both conditions in Lemma 3 are violated.

If the ex-ante constraint is binding, violation of the first condition can be further divided into 2 cases:

Case 1: There exists some with s.t. for every , there exists some s.t. , and . Then, by Lemma 2 there exists some strategy which misreports on and reports truthfully on , which gives higher utility than reporting truthfully without violating the ex-ante constraint, and this contradicts with incentive compatibility.

Case 2: There exists some with s.t. for every , there exists some with . Note that for almost every and the corresponding , we have , otherwise there exists some strategy which misreports on , so that the utility is higher than truthfully reporting, and the ex-ante constraint remains satisfied.

Since equation (5) does not hold for almost every when , there exists some with s.t. for every , there exists some with . Similarly, for almost every and the corresponding , we have .

For any , define and by

 ρ+(r)\coloneqqρ(V+(r)), ρ−(r)\coloneqqρ(V−(r)).

Note that is non-decreasing in , while is non-increasing in . Moreover, the existence of gives , and the existence of gives for some sufficiently large . If for any we have , then there exists s.t. .

If there exists s.t. , define

 r0\coloneqqinf{r|ρ+(r)=0}.

Note that we have . Suppose we have , then for sequence given by , we have and , therefore , and there exists s.t. , which contradicts with the definition of . Suppose , then equation (5) holds for almost every given , thus the second condition is satisfied. Otherwise, implies the existence of some s.t. .

Therefore, if both conditions are not satisfied, we can get with . Define and , both with positive measure, and there exists functions , s.t. by Lemma 2, for changes in utilities and LHS of ex-ante constraint defined as

 ΔU1= Ev∈F1[u(x(h1(v)),v)−c1⋅p(h1(v))] −Ev∈F1[u(x(v),v)−c1⋅p(v)], ΔU2= Ev∈F2[u(x(h2(v)),v)−c1⋅p(h2(v))] −Ev∈F2[u(x(v),v)−c1⋅p(v)], ΔC1= Ev∈F1[f(x(h1(v)),v)−c2⋅p(h1(v))] −Ev∈F1[f(x(v),v)−c2⋅p(v)], ΔC2= Ev∈F2[f(x(h2(v)),v)−c2⋅p(h2(v))] −Ev∈F2[f(x(v),v)−c2⋅p(v)],

we have , and . If , then directly applying on contradicts with incentive compatibility. Otherwise, for s.t. , the change in expected utility by reporting according to w.p. on , according to w.p. on , and otherwise truthfully is given by

 μ1ΔU1+μ2ΔU2 > −μ1r0⋅ΔC1−μ2r0⋅ΔC2 = 0.

This contradicts with incentive compatibility. ∎

Given Lemma 1 - 3, the proof of Theorem 1 is quite straightforward:

###### Proof of Theorem 1.

In fact, each of the two conditions in the theorem is equivalent to a corresponding condition in Lemma 3. We only give proof for equivalence of the second condition, because equivalence of the first condition is almost identical in shape and idea with the standard taxation principle in quasi-linear player case.

Suppose the second condition of the theorem holds. Then, for almost every , equation (5) is naturally satisfied for all , and we have only if the ex-ante constraint is binding. By Lemma 3, is incentive compatible. This proves sufficiency of the condition.

Suppose is incentive compatible by satisfying the second condition of Lemma 3. Then, for almost every , for every s.t. , (5) gives

 (c1+rc2)⋅p(v′)≥(c1+rc2)⋅p(v).

Consider defined by:

 px(q)\coloneqqinfv:x(v)=q(c1+rc2)p(v)

Then, for almost every , we have . By definition of infimum, for almost every , we have . With such a , inequality (5) implies equation (4) for almost every , which finishes our proof. ∎

## 5 Differential properties of incentive compatible mechanisms

For the rest of this paper, we make the following assumption on :

###### Assumption 2.

The boundary of has zero probability, that is, .

Therefore, when taking any expectation over with respect to , it suffices to consider only , the interior of . Note that the assumption implies that .

And we make the following assumptions, that the terms in the player’s utility and constraint are all differentiable:

###### Assumption 3.

For every , the single variable functions and are differentiable at .

Therefore, we can define the vector-valued functions

and by

 ^u(v)\coloneqq∇v′[u(x(v′),v)−c1⋅p(v′)]|v′=v,  ∀v∈˚V

and

 ^f(v)\coloneqq∇v′[f(x(v′),v)−c2⋅p(v′)]|v′=v,  ∀v∈˚V.

The following theorem gives necessary condition of incentive compatibility in terms of the mechanism’s differential properties:

###### Theorem 2.

Under Assumption 1, 2 and 3, the interim rule pair is incentive compatible only if at least one of the following two conditions hold:

1. The ex-ante constraint is binding when reporting truthfully, and for almost every , .

2. There exists constant s.t. for almost every , , and if , the ex-ante constraint is binding when reporting truthfully.

To prove Theorem 2, we first state the following useful lemma.

###### Lemma 4.

Given bivariate function s.t. for any , the single variable function is differentiable at , suppose there exists some with s.t. for any , we have , then for any , there exists s.t .

###### Proof.

For any , since , there exists s.t. the open ball . implies the existence of some unit vector s.t. . Therefore, we can find s.t. . ∎

Then we are ready to prove Theorem 2.

###### Proof of Theorem 2.

Suppose is incentive compatible, but neither of the two conditions hold.

If the constraint is unbinding when reporting truthfully, by Assumption 2, the second condition is violated only if there exists some with s.t. for every , . By Lemma 4 for any there exists s.t. . By Lemma 3 such is not incentive compatible, contradiction.

If the constraint is binding when reporting truthfully, by Assumption 2, the first condition of Theorem 2 is violated only if there exists some with s.t. for every , . By Lemma 4 for any there exists s.t. . Such violates the first condition of Lemma 3. The second condition of Theorem 2 is violated only if for any , there exists some with s.t. for every , . By Lemma 4 for any there exists s.t.

 [u(x(v′),v)−c1⋅p(v′)]−[u(x(v),v)−c1⋅p(v)] (6) > −r⋅[f(x(v′),v)−c2⋅p(v′)]+r⋅[f(x(v),v)−c2⋅p(v)],

which violates the second condition of Lemma 3. Therefore such is not incentive compatible, contradiction.

We conclude that, to make incentive compatible, at least one of the two conditions in Theorem 2 holds. ∎

Theorem 2 gives necessary condition of incentive compatibility in terms of differential properties of the interim rules and functions given by the player model. Using Theorem 2 as a starting point, the next section gives another characterization of incentive compatibility under some additional assumptions, and the characterization is identical in shape with the payment identity in classic mechanism design Myerson (1981); Manelli and Vincent (2007).

## 6 Quasi-linear player model

In this section, we restrict our attention to and which are linear in their second argument:

###### Assumption 4.

There exist functions s.t. for any and any , we have

 u(q,v)=u∗(q)⋅v, f(q,v)=f∗(q)⋅v.

Moreover, the and are differentiable on , and can be continuously extended to .

For technical simplicity, we also include the following assumptions. The first assumption holds for any

which can be expressed by a positive probability density function on

. With these assumptions, the first condition of Theorem 2 is ruled out, and we can refer to or as if they are defined on .

###### Assumption 5.

is connected. Moreover, for any with , is dense in .

###### Assumption 6.

There exists some with s.t. for any , ,

###### Assumption 7.

and are continuous on , and can be continuously extended to .

We define the surrogate outcome function and surrogate utility function , which are central in our last characterization of incentive compatibility.

###### Definition 2.

The surrogate outcome is defined to be:

 ~u(q)\coloneqqu∗(q)+r⋅f∗(q).

The surrogate utility is defined to be:

 ~U(v)\coloneqq~u(x(v))⋅v−(c1+rc2)⋅p(v)

Note that the global constant is a parameter of surrogate outcome and surrogate utility, and such an exists if and only if is incentive compatible. When , the surrogate utility is equal to the player’s original utility. When , the surrogate utility is a linear combination of a utility term and a constraint term, which again mirrors the results in auto-bidding literature.

With the additional assumptions, we are ready to give our second characterization of incentive compatibility.

###### Theorem 3.

Under Assumption 1 - 7, the interim rule pair is incentive compatible if and only only if, for surrogate outcome and surrogate utility given by Definition 2, we have:

• and satisfy AND

• is convex, AND

• if , the ex-ante constraint is binding when reporting truthfully.

###### Proof.

By Theorem 2 and Assumption 6, is incentive compatible only if there exists constant s.t. for almost every , , and if , the ex-ante constraint is binding when reporting truthfully.

By Assumption 4, and can be given by:

 ^u(v)=∇[u∗(x(v))]⋅v−c1⋅∇p(v) ^f(v)=∇[f∗(x(v))]⋅v−c2⋅∇p(v)

For almost every ,