The Optimality of Upgrade Pricing

### 3.2 A Sufficient Condition for Regular Distributions

Our first set of sufficient conditions for upgrade pricing optimality consists of a weak monotonicity condition and a regularity condition.

We call a type distribution $F$ weakly monotone with cutoffs $i^{1}, i^{2}, \ldots, i^{d} \in[n]$ if for any $i, j \in[n]$ and $k \in\{1,2, \ldots, d\}$,

$$
i \leq i^{k} \leq j \Longrightarrow \theta_{i}^{k} \leq \theta_{j}^{k} .
$$

Note that weak monotonicity is strictly weaker than monotonicity: for each item, only order comparisons with respect to a cutoff type need to hold, whereas types above or below the cutoff can be arbitrarily ordered.

Similarly, a type distribution $F$ is regular with respect to cutoffs $i^{1}, i^{2}, \ldots, i^{d} \in$ $[n]$ if for any $i, j \in[n]$ and $k \in\{1,2, \ldots, d\}$,

$$
i \leq i^{k} \leq j \Longrightarrow \phi_{i}^{k} \leq 0 \leq \phi_{j}^{k},
$$

where $\phi_{i}$ denotes the initial $d$-dimensional virtual values

$$
\phi_{i}:=\theta_{i}-\frac{1-F_{i}}{f_{i}}\left(\theta_{i+1}-\theta_{i}\right) .
$$

The initial $d$-dimensional virtual values can be seen as multi-dimensional versions of the virtual values in ?.

We say that a type distribution $F$ is compatibly weakly monotone and regular if it is both weakly monotone and regular with respect to the same set of cutoffs. When such cutoffs $i^{k}$ exist, they are essentially unique except between contiguous types of vanishing virtual value $\phi_{i}^{k}$ and monotone types $\theta_{i}^{k}, i \in[n], k \in[d]$. Subfigure 1 a illustrates a type distribution with this property.

Our regularity condition can be equivalently stated in terms of the pseudorevenues

$$
R_{i}^{k}:=\left(1-F_{i-1}\right) \theta_{i}^{k} .
$$

Subfigure 1 clepicts pseudo-revenues. We call (4) pseudo-revenue because, without an assumption that the values are monotone with respect to the componentwise partial order, the pseudo-revenue does not correspond to the revenue from sales of item $k$ at a posted price of $\theta_{i}^{k}$. In particular, because we have

$$
\begin{aligned}
\frac{R_{i}^{k}-R_{i+1}^{k}}{f_{i}} & =\frac{\left(1-F_{i-1}\right) \theta_{i}^{k}-\left(1-F_{i}\right) \theta_{i+1}^{k}}{f_{i}} \\
& =\frac{f_{i} \theta_{i}^{k}-\left(1-F_{i}\right)\left(\theta_{i+1}^{k}-\theta_{i}^{k}\right)}{f_{i}}=\theta_{i}^{k}-\frac{1-F_{i}}{f_{i}}\left(\theta_{i+1}^{k}-\theta_{i}^{k}\right)=\phi_{i}^{k}
\end{aligned}
$$

imposing regularity with respect to the cutoffs $i^{k}$ is equivalent to requiring that $R_{i}^{k}$ is single-peaked with peak $i^{k}$. While pseudo-revenues do not have immediate economic meaning, they are an important technical tool, in particular for our analysis of non-regular distributions in section 4.

Theorem 1. If the type distribution $F$ is compatibly weakly monotone and regular with respect to cutoffs $\left(i^{k}\right)_{k \in[d]}$, then upgrade pricing is optimal. In particular, the following mechanism is optimal:

$$
q_{i}^{k}:=\left\{\begin{array}{cc}
1 & i \geq i^{k} \\
0 & \text { else. }
\end{array}, \quad i \in[n], k \in[d] .\right.
$$

Proof of Theorem 1. Define the dual variables

$$
\hat{\lambda}_{j i}= \begin{cases}1-F_{i} & \text { if } j=i+1 \\ 0 & \text { else. }\end{cases}
$$

Observe that, by definition, $\hat{\lambda}$ induces the initial virtual values, $\phi_{i}=\phi_{i}^{\hat{\lambda}}$.
We check the properties of Lemma 1. Virtual welfare maximization, condition 2. follows from

$$
q_{i}^{k}=1 \stackrel{(6)}{\Longleftrightarrow} R_{i}^{k} \geq R_{i+1}^{k} \stackrel{(5)}{\Longleftrightarrow} \phi_{i}^{k} \geq 0 .
$$

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Fig. 1: Types, virtual values and pseudo-revenues for type space $\Theta=\left\{\left(\frac{9}{128}, \frac{27}{64}\right),\left(\frac{1}{4}, \frac{3}{2}\right),\left(\frac{1}{2}, 2\right),(1,1)\right\}$ and type distribution $f=\left(\frac{7}{16}, \frac{3}{16}, \frac{1}{8}, \frac{1}{4}\right)$. The optimal mechanism sells good 2 at a price of 1 and good 1 as an upgrade, also at a price of 1 . All types except $\theta_{1}$ buy good 2, and only type $\theta_{4}$ buys good 1.

For flow preservation, condition 3, observe that

$$
\sum_{j=1}^{n} \hat{\lambda}_{i j}-\sum_{j=0}^{n} \hat{\lambda}_{j i}=1-F_{i-1}-\left(1-F_{i}\right)=f_{i} .
$$

The mechanism is implementable, condition 5. by assumption of compatible weak monotonicity and regularity.

Finally, we need to check that complementary slackness (condition 4) holds. Observe that $\hat{\lambda}_{i j}>0$ implies $j=i-1$. Hence, all types must be indifferent between their allocation and payment and the allocation and payment of the next lower type. If the next lower type has the same allocation and payment, this is clearly satisfied. Otherwise, this is the first type buying an upgrade. If this type were not indifferent between buying it and not buying it, the price of the upgrade could be raised, and the revenue increased, without affecting other types' incentives. Thus, this type must be indifferent between their allocation (and payment) and the next lower type's allocation. $\square$

Our assumptions of regularity and weak monotonicity relax the monotonicity of types and Myersonian virtual values by allowing for permutations above and below the monopoly price. These assumptions nonetheless require that the set of types that buy each object remains an upper selection, and conversely the set of types that do not buy remains a lower selection. The intuition for why this works is similar to the idea that the monopoly price does not depend on the valuations of types that are not marginally buying, just as long as they do not become marginal buyers.

These assumptions depend on the fixed order of types we have introduced in the model. Thus, if there exists an order that satisfies these assumptions, upgrade pricing is optimal. Furthermore, multiple orders of types might satisfy the theorem's conditions for a given type distribution $F$. In this case, the theorem can be used to certify optimality of mechanism (6), based on the different orders.

As optimality of a mechanism for a distribution $F$ does not depend on the order on types, the revenue of (6) must be the same for all orders with which the conditions of Theorem 1 are satisfied.

Our next set of conditions imposes similar requirements, strengthened appropriately to allow for non-regular type distributions, which require ironing.


The Optimality of Upgrade Pricing

Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.

Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

## 4 Optimal Mechanisms for Non-Regular Distributions

We now establish the optimality of an upgrade pricing mechanism in settings without regularity. The weaker sufficient conditions will replace the regularity condition and will allow for ironing to be part of the optimal mechanism. The new sufficient conditions will serve to allow us to perform the ironing procedure item-by-item, and limit the interaction of constraints across items. We say that a type space $\Theta$ has monotone marginal rates of substitution if

$$
1 \leq i \leq j \leq n \text { and } 1 \leq k \leq l \leq d \Longrightarrow \frac{\theta_{i}^{l}}{\theta_{i}^{k}} \leq \frac{\theta_{j}^{l}}{\theta_{j}^{k}} .
$$

for any $i, j \in[n], l, k \in[d]$.
Recall that pseudo-revenue is given by $R_{i}^{k}=\left(1-F_{i}\right) \theta_{i}^{k}$.
We call a scalar sequence $\left(R_{i}\right)_{i \in[n]}$, quasi-concave if there is a cutoff $i^{\prime} \in[n]$ such that $i^{\prime} \leq i \leq j$ or $j \leq i \leq i^{\prime}$ implies $R_{i} \geq R_{j}$. We call the point-wise smallest quasi-concave sequence that point-wise dominates $\left(R_{i}\right)_{i \in[n]}$ its quasiconcave closure and denote it by $\left(\bar{R}_{i}\right)_{i \in[n]}$.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Fig. 2: Type space and pseudo-revenues for type space $\Theta=$ $\{(57 / 64,1),(1,5 / 4),(2,3),(9 / 4,5)\}$ and type distribution $f=\left(3 / 8, \frac{1}{4}, \frac{1}{8}, \frac{1}{4}\right)$. The optimal mechanism sells good 1 at a price of $\frac{57}{64}$, and good 2 as an upgrade at a price of 5. All types buy good 1, and only type $\theta_{4}$ buys good 2.

We will make regular use of the sequence $\left(\bar{R}_{i}^{k}\right)_{i \in[n]}$, the quasi-concave closure of the pseudo-revenue for item $k$.

To allow for our construction of a dual certificate of optimality, we need additional assumptions. These will be formulated in terms of candidate ironing
intervals. For a pseudo-revenue $R$, we call a set of contiguous types $I \subseteq[n]$ with

$$
\bar{R}_{i}^{k} \neq R_{i}^{k}
$$

for all $i \in I$ such that there is no superset of contiguous types $I^{\prime} \supseteq I$ such that (8) holds for all $i \in I^{\prime}$, a candidate ironing interval for item $k$. (With slight abuse of language, we refer to discrete sets of contiguous types as intervals.) Every item $k$ may have several candidate ironing intervals, and every type can be contained in a candidate ironing interval for different items.

We relax the regularity assumption on pseudo-revenues $R_{i}^{k}$. Instead of assuming regularity, i.e. $R_{i}^{k}$ to be single-peaked with peak $i^{k}$, we assume two properties that are in combination weaker than regularity. We call a type distribution $F$ mostly regular if for some cutoffs $i^{k} \in \arg \max _{i \in[n]} R_{i}^{k}$ and any $i$ such that $i^{k}<i \leq i^{k+1}$, the following hold:

1. (No partial overlap) If $I$ is a candidate ironing interval of item $k$ and $J$ is a candidate ironing interval of item $k+1$, then either $I \cap J=\emptyset$ and there is $i \in[n]$ such that $I<i<J$ or $J<i<I$, or one of $I, J$ is a subset of the other excluding its endpoints.
2. (No ironing on neighboring maxima) For any ironing candidate interval $I$ of item $k, i^{k}, i^{k+1} \notin I$.
3. (Not too shuffled) For any candidate ironing interval $I \subseteq\left\{i^{k}+1, i^{k}+\right.$ $\left.2, \ldots, i^{k+1}-1\right\}$ and $i \in I$,
$$
\theta_{\min I}^{k+1} \leq \theta_{i}^{k+1} \quad \theta_{\max I}^{k} \leq \theta_{\max I+1}^{k}
$$
Finally, we call a distribution compatibly weakly monotone and mostly regular if it is weakly monotone and mostly regular with respect to the same cutoffs $i^{k}$, $k \in[d]$.

Note that monotone MRS by itself is not a restrictive assumption. For example, in two dimensions, every type set can be ordered in order of monotone MRS. In combination with compatible weak monotonicity and mostly regularity, this assumption becomes stronger.

Subfigure 2a shows the type space of a compatibly weakly monotone and mostly regular type distribution, and subfigure 2b its pseudo-revenues.

Conversely, Figure 3 shows an instance of a distribution over a monotone MRS type space that is not mostly regular. In particular, this example fails the first condition, because it involves overlapping candidate ironing intervals. We will use our assumptions to construct dual variables $\left(\lambda_{i j}\right)_{i, j \in[n]}$ by ironing pseudo-revenues for each item. In our proof that there is an optimal mechanism with an upgrade pricing allocation, we will use monotone MRS to show that for each type, ironing is only needed for two items, the lowest item in the MRS order that the type bought, and the highest item in the MRS order that she didn't buy. We will use the first two conditions of mostly regularity to show that from these two items, we can select a single item to iron at a time, while not changing the other item's virtual values in a way that will break virtual welfare maximization of the allocation. As in Theorem 1 , weak monotonicity ensures implementability

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Fig. 3: Failure of no overlap: $\{2,3\}$ is a candidate ironing interval for item 2, \{3, 4\} is a candidate ironing interval for item 1.

of an upgrade pricing allocation, i.e., the existence of a price vector $\left(t_{i}\right)_{i \in[n]}$ such that the mechanism $(q, t)$ is incentive compatible and individually rational. To allow for our ironing procedure to work, we also need a mild requirement on the monotonicity of types beyond weak monotonicity. While weak monotonicity was a requirement that could be formulated item-by-item, this requirement links the type order of neighboring items.

Theorem 2. Let $\Theta$ have monotone marginal rates of substitution. If the type distribution $F$ is compatibly weakly monotone and mostly regular with respect to cutoffs $\left(i^{k}\right)_{k \in[d]}$, then upgrade pricing is optimal. In particular, the following mechanism is optimal:

$$
q_{i}^{k}:=\left\{\begin{array}{ll}
1 & i \geq i^{k} \\
0 & \text { else },
\end{array} \quad i \in[n], k \in[d] .\right.
$$

Note that the allocation (9) is the allocation that arises from separate monopoly pricing. ${ }^{5}$


The Optimality of Upgrade Pricing

To prove Theorem 2, we will construct a sequence of flows $\lambda_{i}$ from $i=n$ down to $i=1$, starting with $\hat{\lambda}$, the initial flow that induces Myersonian multidimensional virtual values. Given a definition of pseudo-revenue implied by a flow, our Ironing Algorithm will, for each type $i$ and for at least one item $k$, iron to match induced pseudo-revenue with the quasi-concave closure of multidimensional Myersonian pseudo-revenue, $\bar{R}_{i}^{k}$. This is illustrated in Figure 4

The main steps in this proof are to show that the ironing is well-defined in that such implied pseudo-revenue is attainable with a non-negative and feasible flow (Lemma 6 and Lemma 4, respectively). The most technical part of the proof consists of showing that the Ironing Algorithm produces dual variables that maximize virtual welfare (Lemma 3 and Lemma 7 (a)), and satisfy complementary slackness (Lemma 7 (b)).

[^0]Our first lemma is a main structural tool to link different items' virtual values and is tightly connected to monotone MRS. For $k \in[d], i \in[n]$, and flow $\lambda$, denote the normalized virtual value by

$$
\nu_{i}^{\lambda, k}:=\frac{\phi_{i}^{k, \lambda}}{\theta_{i}^{k}}
$$

The property that we will use repeatedly is that $\nu_{i}^{\lambda, k}$ has the same sign as $\phi_{i}^{\lambda, k}$. We call a flow downward if $\lambda_{j i}>0$ for $i, j \in[n]$ implies that $j>i$.

Lemma 2. Let $\Theta$ have monotone MRS. For any non-negative downward flow $\lambda, \nu_{i}^{\lambda, k} \geq \nu_{i}^{\lambda, l}$ for any $1 \leq k \leq l \leq d$ and $i \in[n]$.

Proof. It follows from definitions and monotone marginal rates of substitution that

$$
\begin{aligned}
\frac{\phi_{i}^{\lambda, k}}{\theta_{i}^{k}} & =\frac{\theta_{i}^{k}-\frac{1}{f_{i}} \sum_{j=1}^{n} \lambda_{j i}\left(\theta_{j}^{k}-\theta_{i}^{k}\right)}{\theta_{i}^{k}}=1+\frac{1}{f_{i}} \sum_{j=i}^{n} \lambda_{j i}-\frac{1}{f_{i}} \sum_{j=i}^{n} \lambda_{j i} \frac{\theta_{j}^{k}}{\theta_{i}^{k}} \\
& \geq 1+\frac{1}{f_{i}} \sum_{j=i}^{n} \lambda_{j i}-\frac{1}{f_{i}} \sum_{j=i}^{n} \lambda_{j i} \frac{\theta_{j}^{l}}{\theta_{i}^{l}}=\frac{\theta_{i}^{l}-\frac{1}{f_{i}} \sum_{j=1}^{n} \lambda_{j i}\left(\theta_{j}^{l}-\theta_{i}^{l}\right)}{\theta_{i}^{l}}=\frac{\phi_{i}^{\lambda, l}}{\theta_{i}^{l}} .
\end{aligned}
$$ $\square$

The next Lemma shows that virtual welfare maximization reduces to virtual welfare maximization for the neighboring items, i.e., the last item that a type buys and the first item that a type does not buy-with respect to the MRS order.

Lemma 3. Assume $\Theta$ has monotone MRS and mostly regular and that there exists a non-negative downward flow $\lambda$ such that for any $i \in[n]$ such that $i^{k} \leq$ $i \leq i^{k+1}$, we have $\phi_{i}^{\lambda, k} \geq 0$ and $\phi_{i}^{\lambda, k+1} \leq 0$. Then, the allocation in (9) maximizes virtual welfare.

Proof. Fix $i \in[n]$ such that $i^{k} \leq i \leq i^{k+1}$. Note that as $\phi_{i}^{\lambda, k}$ and $\nu_{i}^{\lambda, k}$ are positive multiples of each other, Lemma 2 implies the implications

$$
\begin{aligned}
\phi_{i}^{\lambda, k+1} \leq 0 & \Longrightarrow \phi_{i}^{\lambda, l} \leq 0, \quad l \geq k+1 \\
\phi_{i}^{\lambda, k} \geq 0 & \Longrightarrow \phi_{i}^{\lambda, l} \geq 0, \quad l<k .
\end{aligned}
$$

Therefore, the assumption implies that $\phi_{i}^{\lambda, l} \leq 0$ for any $l>k$ and $\phi_{i}^{\lambda, l} \geq 0$ for any $l \leq k$, which ensures virtual welfare maximization of (9). $\square$

For $k=0$ and $k=d$ this Lemma reduces virtual welfare maximization for all items, and ironing for all items, to virtual welfare maximization for the first resp. last item. Finding a flow that maximizes virtual welfare reduces to ironing the (one-dimensional) virtual values $\phi_{i}^{1}$ and $\phi_{i}^{d}$. For types $i \leq i^{1}$ and $i \geq i^{d}$,
we can hence use techniques from one-dimensional ironing and iron the pseudorevenue to its concave closure in a discrete variant of ?'s procedure. From now, our discussion therefore focuses on $k \in[d-1]$ and $i \in\left[i^{k}+1, i^{k+1}\right]$, i.e. types where an ironing that ensures virtual welfare maximization for both item $k$ and item $k+1$ is needed.

The following algorithm will make use of $\hat{\lambda}$ as defined in (7), the initial flow and of a generalization of the pseudo-revenue. The pseudo-revenue associated to a flow $\lambda, R_{i}^{\lambda, k}$ is

$$
R_{i}^{\lambda, k}=\sum_{j=i}^{n} f_{j} \phi_{j}^{\lambda, k}
$$

This generalization is intuitive, as virtual values are, as in (5), slopes of pseudorevenues

$$
\frac{R_{i}^{\lambda, k}-R_{i+1}^{\lambda, k}}{f_{i}}=\frac{\sum_{j=i}^{n} f_{j} \phi_{j}^{\lambda, k}-\sum_{j=i+1}^{n} f_{j} \phi_{j}^{\lambda, k}}{f_{i}}=\phi_{i}^{\lambda, k} .
$$

Our algorithm will adjust a flow by raising one point in a revenue sequence at a time, from right to left. We will prove that this will yield slopes of revenue sequences-i.e. virtual values-which have the correct sign for virtual welfare maximization of (9). This is non-trivial, as pseudo-revenues for different items might not move in the same direction when dual variables are changed.

$$
\begin{aligned}
& \lambda \leftarrow \hat{\lambda} ; \\
& \text { for } i=n \text { to } 1 \text { do } \\
& \qquad \begin{array}{l}
\text { Let } \gamma_{i} \in[0,1] \text { be maximal such that for } \\
\lambda_{j i}^{\prime} \leftarrow \gamma_{i} \lambda_{j i}, \quad \forall j: n>j>i \\
\lambda_{j(i-1)}^{\prime} \leftarrow \lambda_{j(i-1)}+\left(1-\gamma_{i}\right) \lambda_{j i}, \quad \forall j: n>j>i \\
\lambda_{i(i-1)}^{\prime} \leftarrow \lambda_{i(i-1)}-\left(1-\gamma_{i}\right) \sum_{i^{\prime}=i}^{n} \lambda_{i^{\prime} i} \\
R_{i}^{\lambda^{\prime}, \kappa(i)}=\bar{R}_{i}^{\kappa(i)} \text { holds; } \\
\lambda \leftarrow \lambda^{\prime} ;
\end{array}
\end{aligned}
$$

Return $\lambda^{\prime}$;
Algorithm: Ironing, parameterized by an ironing mapping $\kappa:[n] \rightarrow[d]$
The flow (11) was used earlier in Haghpanah and Hartline (2020). An important difference is that Haghpanah and Hartline (2020) choose $\gamma_{i}$ to iron the revenue sequence of the grand bundle to the concave closure of pseudo-revenue. Instead, we iron to the quasi-concave closure of (their equivalent of) pseudorevenue of an item $\kappa(i)$. The parameter $\gamma_{i}$ can be found as solution to a system of linear equations. We show that a solution $\gamma_{i} \in[0,1]$ exists in Lemma 6.


The Optimality of Upgrade Pricing

We first observe that the Ironing Algorithm outputs a flow which is nonnegative and feasible.

Lemma 4. The output of the Ironing Algorithm is a flow, i.e. non-negative and satisfies flow preservation, Lemma 1 item 3.

Proof. We prove the claim by induction from $i=n$ to $i=1$. Note that $\hat{\lambda}$ is feasible as argued in the proof of Theorem 1, which starts the induction. Fix an arbitrary iteration $i \in[n]$, and assume that $\lambda$ is feasible. We check that the difference in excess flow, i.e. incoming and outgoing flow, cancel out for $j>i$. For $i$ and $i+1$ similar calculations yield the result. We omit these. For any $j>i$,

$$
\lambda_{j i}^{\prime}-\lambda_{j i}+\lambda_{j(i-1)}^{\prime}-\lambda_{j(i-1)}=\gamma \lambda_{j i}-\lambda_{j i}+\lambda_{j(i-1)}+\left(1-\gamma_{i}\right) \lambda_{j i}-\lambda_{j(i-1)}=0 .
$$

Now consider non-negativity. Each $\lambda_{i j}$ reduces at most once during the course of the Ironing Algorithm. More specifically, only if $j=i-1$ and during iteration $i$. In this iteration,

$$
\lambda_{i(i-1)}^{\prime}=\lambda_{i(i-1)}-\left(1-\gamma_{i}\right) \sum_{i^{\prime}=i}^{n} \lambda_{i^{\prime} i} \geq \lambda_{i(i-1)}-\sum_{i^{\prime}=i}^{n} \lambda_{i^{\prime} i}=f_{i} \geq 0,
$$

where we used that $\gamma_{i} \leq 1, \lambda_{i r}=\hat{\lambda}_{i r}, r<i-1$, which in particular implies that $\lambda_{i r}=0$, and feasibility of the flow. $\square$

Next observe that in the Ironing Algorithm, iteration $i$ changes the revenue (for any item $k$ ) only for type $i$. Hence, our ironing algorithm raises pseudorevenue for one type at a time.

Lemma 5. For any iteration $i, R_{j}^{\lambda^{\prime}, k}=R_{j}^{\lambda, k}$ for any $j \neq i$. In particular, $\phi_{j}^{\lambda^{\prime}, k}=\phi_{j}^{\lambda, k}$ for $j \notin\{i-1, i\}$.

Proof. First note that as the in-flow for higher types remains unchanged in iteration $i, \lambda_{r j}^{\prime}=\lambda_{r j}, r \in[n], j>i$, the revenue does not change, $R_{j}^{\lambda^{\prime}, k}=R_{j}^{\lambda, k}$. For types $j<i$, we check that the changes to virtual welfare on the types whose inflows do change, $i$ and $i-1$, cancel out. By definition of virtual values,

$$
\begin{aligned}
\phi_{i}^{\lambda^{\prime}, k} & =\phi_{i}^{\lambda, k}+\frac{1-\gamma_{i}}{f_{i}} \sum_{j=i}^{n} \lambda_{j i}\left(\theta_{j}^{k}-\theta_{i-1}^{k}\right) \\
\phi_{i-1}^{\lambda^{\prime}, k} & =\phi_{i-1}^{\lambda, k}+\frac{1-\gamma_{i}}{f_{i-1}} \sum_{j=i}^{n} \lambda_{j i}\left(\theta_{j}^{k}-\theta_{i-1}^{k}\right)-\frac{1-\gamma_{i}}{f_{i-1}} \sum_{j=i}^{n} \lambda_{j i}\left(\theta_{j}^{k}-\theta_{i-1}^{k}\right) \\
& =\phi_{i}^{\lambda, k}-\frac{1-\gamma_{i}}{f_{i}} \sum_{j=i}^{n} \lambda_{j i}\left(\theta_{j}^{k}-\theta_{i-1}^{k}\right) .
\end{aligned}
$$

Hence,

$$
f_{i-1} \phi_{i-1}^{\lambda^{\prime}, k}+f_{i} \phi_{i}^{\lambda^{\prime}, k}=f_{i-1} \phi_{i-1}^{\lambda, k}+f_{i} \phi_{i}^{\lambda, k}
$$

The statement on the virtual values follows from (10). $\square$

Before showing that $\gamma_{i}$ in the algorithm always exists, we define the ironing function $\kappa(i)$.

By no ironing on neighboring maxima, each candidate ironing interval $I$ must be contained in an interval $\left\{i^{k}, i^{k}+1, \ldots, i^{k+1}\right\}$. By this condition, in addition to
no partial overlap, for each type $i$, there is a unique inclusion maximal candidate among the candidate ironing intervals for items $k$ and $k+1$. We let $\kappa(i)$ denote the item this interval is a candidate ironing interval for. If $i$ is not part of any ironing interval, we set $\kappa(i)$ arbitrarily in $\{k, k+1\}$. We call $\kappa(i)$ the ironed item for type $i$ and piece-wise constant intervals of $\kappa$ ironing intervals.

Lemma 6. Assume that $F$ is mostly regular. Then, for each $i \in[n], \gamma_{i}$ such that $R_{i}^{\lambda_{i}\left(\gamma_{i}\right), \kappa(i)}=\bar{R}_{i}^{\kappa(i)}$ exists. In particular, the Ironing Algorithm is well-defined.

Proof. We prove this statement by induction from $i=n$ down to 1 . Let $i \in[n]$ and assume that $R_{i+1}^{\lambda, \kappa(i)}=\bar{R}_{i+1}^{\lambda, \kappa(i)}$. If $i$ is not part of an ironing interval, then by definition of ironing intervals and Lemma $5, R_{i}^{\lambda, k}=\bar{R}_{i}^{\lambda, k}$, and the induction step is trivial by choosing $\gamma_{i}=1$, yielding $R_{i}^{\lambda^{\prime}(1), k}=\bar{R}_{i}^{\lambda^{\prime}(1), k}$. Otherwise, $i$ is in an ironing interval. Let $\kappa(i)=k$. By no partial overlap, if $i+1$ is part of an ironing interval, it must be part of the same ironing interval, in particular must have been ironed for item $k$. Hence, by the induction hypothesis, $R_{i+1}^{\lambda, k}=\bar{R}_{i+1}^{\lambda, k}$.

Denote

$$
\bar{\phi}_{i}^{k}=\frac{\bar{R}_{i}^{k}-\bar{R}_{i+1}^{k}}{f_{i}}
$$

the slope of the quasi-concave closure of pseudo-revenue of item $k$ at type $i$. By definition of the quasi-concave closure, the slope of the revenue curve must be non-positive,

$$
\bar{\phi}_{i}^{k} \leq 0 .
$$

As all types are non-negative, we get that

$$
\overline{\phi_{i}^{k}} \leq 0 \leq \theta_{i}^{k}=\phi_{i}^{\lambda^{\prime}(0), k} \text {. }
$$

Again by Lemma 5, $\bar{R}_{i+1}^{k}=R_{i+1}^{\lambda^{\prime}(0), k}$. Therefore

$$
\begin{aligned}
\bar{R}_{i}^{k} & =f_{i} \overline{\phi_{i}^{k}}+\bar{R}_{i+1}^{k}=f_{i} \overline{\phi_{i}^{k}}+R_{i+1}^{\lambda^{\prime}(0), k} \\
& \leq f_{i} \phi_{i}^{\lambda^{\prime}(0), k}+R_{i+1}^{\lambda^{\prime}(0), k}=R_{i}^{\lambda_{i}(0), k}
\end{aligned}
$$

In particular, $\bar{R}_{i}^{k} \leq R_{i}^{\lambda_{i}(0), k}$.
Also, by Lemma 5 and the definition of the quasi-concave closure, $R_{i}^{\lambda^{\prime}(1), k}=$ $R_{i}^{\lambda, k} \leq \bar{R}_{i}^{k}$. As $\gamma \mapsto R_{i}^{\lambda^{\prime}}(\gamma), k$ is a continuous function, the existence of the desired $\gamma \in[0,1]$ follows from the Intermediate Value Theorem. $\square$

The last lemma before the proof of Theorem 2 shows that the output of the algorithm satisfies complementary slackness and the condition of Lemma 3, which is sufficient for virtual welfare maximization.

Lemma 7. Assume that $\Theta$ is has monotone MRS, and that $F$ is mostly regular. Then, $q$ maximizes virtual welfare and satisfies the requirements of Lemma 3 with respect to $\lambda^{\prime}$, the output of the Ironing Algorithm.


The Optimality of Upgrade Pricing

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Fig. 4: Ironing of virtual values and corresponding pseudo-revenues.

Proof of Lemma 7. We first show that $\lambda^{\prime}$ satisfies the requirements of Lemma 7, i.e. that for any $i^{k} \leq i \leq i^{k+1}$ we have that $\phi_{i}^{\lambda, k} \geq 0$ and $\phi_{i}^{k+1, \lambda} \leq 0$. Let $i \in[n]$. If $i$ is not in an ironing interval, and there is no ironing interval $I$ such that $i=\min I-1$, the claim follows from the definition of the quasi-concave closure and our definition of ironing intervals, as well as Lemma 5.

We consider the remaining cases $i=\max I, i=\min I-1$, and $i \in I \backslash\{\max I\}$ separately. For the third case, by definition of the ironing and the quasi-concave closure,

$$
\phi_{i}^{\lambda^{\prime}, \kappa(i)}=\nu_{i}^{\lambda^{\prime}, \kappa(i)}=0 .
$$

For $\kappa(i)=k, \nu_{i}^{k+1, \lambda} \leq 0$ by Lemma 2 and hence $\phi_{i}^{k+1, \lambda} \leq 0$ as $\nu_{i}^{k+1, \lambda}$ and $\phi_{i}^{k+1, \lambda}$ are positive multiples of each other. Similarly, we have for $\kappa(i)=k+1$ that $\nu_{i}^{k, \lambda} \geq 0$ by Lemma 2 and hence $\phi_{i}^{k, \lambda} \geq 0$. It remains to consider $i=\min I-1$ and $i=\max I$. We consider these separately for $\kappa(i)=k$ and $\kappa(i)=k+1$, for a total of four cases.

Case 1. $i=\max I$ and $\kappa(i)=k$. By definition of the quasi-concave closure, $\bar{R}_{i}^{k}=R_{i}^{k}$, the algorithm chooses $\gamma_{i}=1$ and hence $\phi_{i}^{\lambda^{\prime}, k}=\phi_{i}^{k}=0$ and hence by arguments as above, $\phi_{i}^{\lambda^{\prime}, k+1}=\phi_{i}^{k+1} \leq 0$.

Case 2. $i=\min I-1$ and $\kappa(i)=k+1$. By definition of the quasi-concave closure, $\overline{R_{i}^{k+1}}=R_{i}^{k+1}$, the algorithm chooses $\gamma_{i}=1$ and hence, by an argument similar to case $1, \phi_{i}^{\lambda, k}=\phi_{i}^{k} \geq 0$.

Case 3. $i=\max I$ and $\kappa(i)=k+1$.
The derivative of the virtual value of the right end of the ironing interval, type $i$ is given by

$$
\frac{\partial \phi_{i}^{\lambda^{\prime}, k}}{\partial \gamma}=-\frac{1}{f_{i}} \sum_{j=i+1}^{n} \lambda_{j i}\left(\theta_{j}^{k}-\theta_{i}^{k}\right)=-\frac{1}{f_{i}} \lambda_{(i+1) i}\left(\theta_{i+1}^{k}-\theta_{i}^{k}\right) .
$$

By no partial overlap, $0 \leq \phi_{i}^{k}$. Also, by not too shuffledness, (13) is nonpositive. Hence, $\phi_{i}^{k}=\phi_{i}^{\gamma^{\prime}(0), k} \leq \phi_{i}^{\lambda^{\prime}, k}$, and the algorithm chooses $\gamma_{i}<1$. Combining these observations, we obtain $\phi_{i}^{k, \lambda^{\prime}} \geq 0$.

Because the algorithm chooses $\gamma_{i}<1$, this implies

$$
\phi_{i}^{\lambda^{\prime}, k} \geq \phi_{i}^{k} \geq 0 .
$$

Case 4. $i=\min I-1$ and $\kappa(i)=k$. The derivative of the virtual value of the next item at the left end of the ironing interval, type $i$ is given by

$$
\begin{aligned}
\frac{\partial \phi_{i}^{\lambda^{\prime}, k+1}}{\partial \gamma} & =\frac{1}{f_{i}} \sum_{j=i+1}^{n} \lambda_{j i}\left(\theta_{j}^{k+1}-\theta_{i}^{k+1}\right)-\frac{1}{f_{i}} \sum_{j=i+1}^{n} \lambda_{j i}\left(\theta_{i+1}^{k+1}-\theta_{i}^{k+1}\right) \\
& =\frac{1}{f_{i}} \sum_{j=i+1}^{n} \lambda_{j i}\left(\theta_{j}^{k+1}-\theta_{i+1}^{k+1}\right) \geq 0,
\end{aligned}
$$

where the last inequality follows from not-too-shuffledness (note that $i+1=$ $\min I)$. By no partial overlap, $0 \geq \phi_{i}^{k+1}$. Moreover, $\phi_{i}^{k+1}=\phi_{i}^{\gamma^{\prime}(0), k} \geq \phi_{i}^{\lambda^{\prime}, k+1}$ because of (14) and because the algorithm chooses $\gamma_{i}<1$. Combining these observations, we obtain $\phi_{i}^{k+1, \lambda} \leq 0$.

To show complementary slackness, observe that whenever $i$ is not in an ironing interval, as $\gamma_{i}$ is chosen maximal such that $\bar{R}_{i}^{\kappa(i)}=R_{i}^{\kappa(i)}$, the Algorithm chooses $\gamma_{i}=1$, which implies that for $j>i>r$ and for $j>i+1$ and $i=r$, $\lambda_{j r}^{\prime}=0$. By no ironing over maxima, this implies that for $j>i^{k}>r, \lambda_{j r}=0$. Moreover, as an invariant of the algorithm the flow $\lambda$ is downward. Hence, the only dual variables that are tight are within types that get the same allocation and payment (corresponding to $\lambda_{i j}$ such that $i^{k} \leq i, j \leq i^{k+1}, k \in[d]$ ) or local downward constraints (corresponding to $\lambda_{(i+1) i}, i \in[n-1]$ ). The former incentive constraints clearly bind, the latter bind by weak monotonicity, as for the marginally buying type (which, by compatibility of weak monotonicity with mostly regularity must also be the first type in the MRS order), the price could be raised if she were not indifferent between her allocation and payment and the allocation and payment of the next lower type. $\square$

Having this result, we are ready to finish the proof of Theorem 2.
Proof of Theorem 2. Implementability follows from weak monotonicity and the definition of the optimal mechanism, (9). Non-negativity and feasibility of flow are properties of the Ironing Algorithm shown in Lemma 4. Virtual welfare maximization and complementary slackness have been shown in Lemma 7. $\square$


The Optimality of Upgrade Pricing

## 5 Upgrade Pricing and Separate Pricing

In both Theorem 1 and Theorem 2, we established the optimality of an upgrade pricing mechanism that yields the same allocation as separate (item by item) monopoly pricing, though not necessarily the same transfers. We will show in this section that, under monotonicity with respect to the component-wise partial order, separate pricing and upgrades become equivalent-upgrade pricing is redundant.

We say that the type space $\Theta$ is monotone if $\theta_{i}^{k} \leq \theta_{j}^{k}$ for any $i<j \in[n]$ and $k \in[d]$.

We call a mechanism separate pricing if a type separately chooses whether to buy each item $k$ at a price $p_{k}$. Formally, a mechanism satisfies separate pricing if it can be written as:

$$
q_{i}^{k}=\left\{\begin{array}{ll}
1 & \theta_{i}^{k} \geq p_{k} \\
0 & \text { else, }
\end{array} \quad t_{i}=\sum_{k=1}^{d} p_{k} \mathbb{1}_{q_{i}^{k}=1}\right.
$$

Theorem 3. If the type space $\Theta$ is monotone, then the outcome of any upgrade pricing mechanism can be implemented via separate pricing, and conversely. When the type space is not monotone, neither implication needs to hold.

Proof. We first assume types are monotone and show that the allocation and revenue of any upgrade pricing mechanism can be obtained through a separate pricing mechanism, and vice versa.

Let $\theta_{1} \leq \cdots \leq \theta_{i} \leq \cdots \leq \theta_{n}$, and fix an upgrade pricing mechanism $\mathcal{M}$. This mechanism admits an indirect representation as (a) a collection of bundles ranked by set inclusion $\left\{b_{k}\right\}_{k=0}^{K}$, with $b_{0}=\varnothing$ and $K \leq d$, and (b) a vector of prices $t_{k}$ that are increasing in $k$, with $t_{0}=0$. Let $\underline{\theta}_{k}$ and $\bar{\theta}_{k}$ denote the lowest and highest types who choose bundle $b_{k}$ under mechanism $\mathcal{M}$. Because types are monotone, buyer self-selection implies $\underline{\theta}_{k} \geq \bar{\theta}_{k-1}$.

We now construct a separate pricing mechanism, i.e., a vector of prices $\left\{p_{j}\right\}_{j=1}^{d}$ that yields the same allocation and payments as our upgrade pricing mechanism. To do so, define the collection of upgrades $u_{k}:=b_{k} \backslash b_{k-1}$ and the upgrade prices $\tau_{k}:=t_{k}-t_{k-1}$. For each upgrade bundle $k$ and every good $j \in u_{k}$, let the single-item prices $p_{j}$ satisfy

$$
p_{j} \in\left[\bar{\theta}_{k-1}^{j}, \underline{\theta}_{k}^{j}\right] \quad \text { and } \quad \sum_{j \in u_{k}} p_{j}=\tau_{k} .
$$

Under monotonicity, such a vector of prices always exists. By consumer selfselection in the original mechanism $\mathcal{M}$, we have

$$
\begin{aligned}
\bar{\theta}_{k-1} b_{k}-t_{k} & \leq \bar{\theta}_{k-1} b_{k-1}-t_{k-1}, \\
\underline{\theta}_{k} b_{k-1}-t_{k-1} & \leq \underline{\theta}_{k} b_{k}-t_{k} .
\end{aligned}
$$

In turn, this implies

$$
\bar{\theta}_{k-1} u_{k} \leq \tau_{k} \leq \underline{\theta}_{k} u_{k} .
$$

With the prices so constructed, each type purchases the same goods as under $\mathcal{M}$ and pays the same total price. Notice first that each type's choice from the original mechanism $\mathcal{M}$ is still available at the same price, i.e., each bundle $b_{k}$ can still be purchased for a total price $t_{k}$. Moreover, by monotonicity, no type $\theta$ who buys bundle $b_{k}$ under the upgrade pricing mechanism $\mathcal{M}$ derives positive net surplus from any object $j \in u_{k^{\prime}}$ with $k^{\prime}>k$ under the separate prices
constructed above. And finally, no such type $\theta$ derives positive net surplus by removing any object $j \in u_{k^{\prime}}$ with $k^{\prime} \leq k$ from her consumption bundle.

The other direction of this result is immediate: if types are monotone, the goods purchased by two different types under any separate pricing mechanism are ranked by set inclusion. Thus, replacing the separate pricing mechanism with the resulting upgrade pricing mechanism yields the same outcome.

Finally, we show by means of two counterexamples that, without type monotonicity, separate pricing is not equivalent to upgrade pricing.

In particular, there exist type spaces and vectors of separate prices that do not induce an upgrade pricing allocation. For example, let

$$
\Theta=\{(1,1),(1,3),(3,3),(4,1)\}
$$

and consider the separate prices $p=(2,2)$ : type $\theta_{2}$ buys good 2 only, type $\theta_{3}$ buys both goods, and type $\theta_{4}$ buys good 1 only.

Likewise, for the same type space, consider the upgrade pricing mechanism where $q=(0,1)$ is sold for $t=2$ and $q=(1,1)$ is sold for $t=4$, i.e., good $j=1$ is only sold as an upgrade, for an additional price $\tau=2$. Under this mechanism, type $\theta_{2}$ buys good 2 only, while types $\theta_{3}$ and type $\theta_{4}$ buy both goods. However, as we saw above, the vector of separate prices $p=(2,2)$ yields a different allocation (and a lower revenue for the seller). $\square$

Whenever an upgrade pricing mechanism implements the allocation of optimal separate pricing, each marginal type $\underline{\theta}_{k}$ is indifferent by construction between the two consecutive bundles $b_{k-1}$ and $b_{k}$. Theorem 3 then implies that the outcome of this mechanism can be implemented by the separate monopoly prices.

Corollary 1. If $\Theta$ is monotone, $q$ is an allocation of an optimal upgrade pricing mechanism, and $q$ is the allocation of separate monopoly pricing, then separate monopoly pricing is optimal.

Adding a monotonicity condition to both of our main theorems, Theorem 1 and Theorem 2, we hence obtain two sets of sufficient conditions under which separate monopoly pricing is optimal.

Corollary 2. If $\Theta$ is monotone and $F$ is regular, separate monopoly pricing is optimal.

Corollary 3. If $\Theta$ is monotone and has a monotone marginal rates of substitution, and $F$ is mostly regular, then separate monopoly pricing is optimal.


The Optimality of Upgrade Pricing

## 6 Conclusion

It is a common practice for a seller to offer bundles of products or services that are ordered in a way that more expensive bundles contain all items from less expensive bundles as well as some extra items. In this paper, we provide
sufficient conditions under which such "upgrade pricing" schemes are exactly optimal for a monopolist seller.

There are several ways in which the current analysis could be extended. First, our conditions could be relaxed to account for richer type spaces and type distributions, such as a continuum of types in the d-dimensional space. One natural extension can be obtained immediately: assume that a type distribution can be split into several type cohorts, in fact quantized type space, such that each type cohort satisfies the conditions of our theorems. Our results imply that the optimal mechanisms in each respective cohort are upgrade pricing. In this respect, Bergemann et al. (2021) show that in nonlinear pricing problems, the revenue of the continuous type space is generally well approximated by a finite quantized type space.

Second, our sufficient conditions for the optimality of upgrade pricing may be complemented by necessary conditions. In doing so, one may want to distinguish between conditions on type distributions and type spaces. For example, one may ask which type spaces guarantee that upgrade pricing is optimal irrespective of the type distribution.

Finally, throughout the paper we highlight the interplay between optimality of different pricing schemes: bundling, upgrade pricing, and separate sales. It would be instructive to provide a more complete characterization of the cases in which one of these schemes strictly outperforms another.

Acknowledgements We thank Mark Armstrong and the seminar audience at MIT for helpful comments. Bergemann and Bonatti acknowledge financial support through NSF SES 1948336. Smolin acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010).


The Optimality of Upgrade Pricing

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J. Philips. Don't look now, but the great unbundling has spun into reverse. New York Times, 2017.

[^0]:    ${ }^{5}$ In section 5, we further explore the relationship between upgrade pricing and separate pricing, by showing conditions under which the allocation (9) can be implemented by a vector of single-item prices.


The Optimality of Upgrade Pricing

## Citation and provenance

**Authors:** Dirk Bergemann; Alessandro Bonatti; Andreas Haupt; Alex Smolin

**Canonical citation:** Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md

**Source record:** https://arxiv.org/abs/2107.10323

**Published record:** https://doi.org/10.1007/978-3-030-94676-0_3

**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

Provenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt


> Machine-readable author manuscript.
> Authors: Laura Doval; Alex Smolin.
> Canonical citation: Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


# The Welfare Impact of Recommendation Algorithms

**Authors:** Laura Doval; Alex Smolin

**Manuscript date:** 2025

#### Abstract

In this letter, we summarize our recent work on the welfare impact of recommendation algorithms and propose questions for further study. We model recommendation algorithms as an information structure, which shapes how a third party takes actions that affect the welfare of different individuals in a population. Each recommendation algorithm thus induces a welfare profile, describing the expected payoffs of different individuals when the third party takes actions following the algorithm. Our framework allows us to characterize and compute the set of all such profiles, which we dub the Bayes welfare set. The Bayes welfare set allows us to reduce society's choice of an algorithm to the choice of a Bayes welfare profile. Our framework complements that of the algorithmic fairness literature which remains agnostic about the population's payoffs, focusing instead on statistical properties of algorithms, such as accuracy, parity, or fairness.

Categories and Subject Descriptors: [Social and Behavioral Sciences]: Economics-
General Terms: Economics, Theory
Additional Key Words and Phrases: Recommendation algorithms, fairness, persuasion, information structures


## 1. INTRODUCTION

Information has increasingly become a tool for shaping society's choices in highstakes domains. While this phenomenon is not exclusive to the "big data" economy-consider the role of scores and ratings in school placement, promotions, and credit allocation-the rise of algorithmic recommendations has highlighted society's growing reliance on information in policy-relevant domains. Consider, for instance, the role of algorithms in recommending who obtains bail [Angwin et al. 2016], credit [Jagtiani and Lemieux 2019], who is hired [Raghavan et al. 2020; Li et al. 2020], and which health treatments to prescribe [Obermeyer et al. 2019], and more recently, generative AI models, which users can leverage as virtual consultants [Immorlica et al. 2024].

The ever-increasing role of recommendation algorithms in high-stakes domains and their obvious welfare impact has caught the attention of the Computer Science and Economics communities. For instance, [Kearns and Roth 2019; 2020] underscore the importance of having portable definitions of privacy or fairness, which can be coupled with the training model's objective, to produce algorithms with desirable outcomes. Whereas the literature on differential privacy and algorithmic fairness is agnostic about how to measure individuals well-being or the objective of the algorithm designer, [Mullainathan 2018; Kleinberg et al. 2018; Rambachan et al. 2020] argue for letting the social planner's objective determine the properties of algorithms.

In [Doval and Smolin 2021; 2024], we provide a framework to study the welfare impact of recommendation algorithms on a population of heterogeneous individuals. Our framework marries welfare economics and information design. It integrates welfare economics because a primitive of our environment is a measure of individual welfare, which could represent the actual utility function of individuals in the society, or the social planner's perception of this utility. It also draws from information design because recommendation algorithms fundamentally operate as information structures, which provide noisy signals about an underlying state of the world to a decision maker who ultimately takes actions on behalf of the individuals. As such, recommendation algorithms are inherently bounded in their ability to generate welfare, the same way an information designer is bounded in their ability to persuade a receiver to take a given action [Dughmi 2017].

Our primary goal in this note is to introduce the readers to our framework, based on illustrations of the results in [Doval and Smolin 2021] and [Doval and Smolin 2024]. Section 2 introduces the simplest version of our framework to lay down the concepts in the simplest terms. In Section 3, we extend the framework so that it is closest to that in algorithmic fairness. We conclude by pointing out applications of our framework to information design and directions for future research at the intersection of Computer Science and Economics.


## 2. BASIC FRAMEWORK

In the basic model, a unit mass population of individuals have types in a finite set $\Theta=\left\{\theta_{1}, \ldots, \theta_{N}\right\}$, drawn from a full support prior distribution, $\mu_{0} \in \Delta(\Theta)$. Each individual's welfare depends on her type $\theta$ and an (unmodeled) outside observer's
belief about her type. We represent this by a welfare function $w: \Delta(\Theta) \times \Theta \rightarrow \mathbb{R}$, representing for each belief $\mu$ and type $\theta$, the welfare of individuals of type $\theta$ when the outside observer's belief is $\mu$. For instance, if individuals' welfare depends on the actions of the outside observer, the welfare function captures in reducedform how the outside observer's action, and hence welfare, changes as the outside observer's beliefs about $\Theta$ changes. Alternatively, the welfare function may capture that the population's welfare may be driven by image or reputation concerns, like in [Bénabou and Tirole 2006], or psychological motives, as in [Lipnowski and Mathevet 2018].

We model algorithms as information structures. An information structure $\Pi=$ $(\pi, S)$ consists of a countable set of labels $S$, and a mapping $\pi$, which associates to each type $\theta$ a distribution over signals $\pi(\cdot \mid \theta) \in \Delta(S)$. Let $\mu_{s}$ denote the posterior belief given signal $s \in S$. An information structure induces two kinds of distribution over posterior beliefs $\left\{\mu_{s}: s \in S\right\}$. First, for each $\theta$, the signal distribution $\pi(\cdot \mid \theta)$ induces a distribution over posterior beliefs conditional on an individual's type being $\theta$. Second, the prior $\mu_{0}$ and the signal distribution induce an unconditional distribution over posterior beliefs. We denote them by $\langle\Pi \mid \theta\rangle$ and $\langle\Pi\rangle$, respectively.

The welfare function $w$ together with an information structure, $\Pi$, defines a welfare profile, $w_{\Pi}: \Theta \mapsto \mathbb{R}$, as

$$
w_{\Pi}(\theta)=\mathbb{E}_{\langle\Pi \mid \theta\rangle}[w(\mu, \theta)]=\sum_{s \in S} \pi(s \mid \theta) w\left(\mu_{s}, \theta\right) .
$$

We denote such a profile, a Bayes welfare profile, and the set of all Bayes welfare profiles, the Bayes welfare set. Formally, the Bayes welfare set is defined as:

$$
\mathrm{W} \equiv\left\{\mathrm{w} \in \mathbb{R}^{N}: \exists \Pi \text { s.t. } \mathrm{w}_{i}=w_{\Pi}\left(\theta_{i}\right) \forall i \in\{1, \ldots, N\}\right\} .
$$

From the point of view of welfare economics, the Bayes welfare set admits a classical interpretation: It is the utility possibility set in an economy in which information structures take the role of allocations.

An apparent difficulty when characterizing the Bayes welfare set is that the Bayes welfare profiles depend on the conditional distributions over posterior beliefs induced by the information structure (cf. Equation (1)). However, we show any Bayes welfare profile satisfies the following:

$$
w_{\Pi}(\theta)=\mathbb{E}_{\langle\Pi \mid \theta\rangle}[w(\mu, \theta)]=\mathbb{E}_{\langle\Pi\rangle}\left[\frac{\mu(\theta)}{\mu_{0}(\theta)} w(\mu, \theta)\right]=\mathbb{E}_{\langle\Pi\rangle}[\hat{w}(\mu, \theta)] .
$$

That is, the expectation of $w$ under $\Pi$ conditional on $\theta$ can be expressed as the unconditional expectation of the truth-adjusted welfare function, $\hat{w}$, under $\Pi$. The truth-adjusted welfare function, $\hat{w}$, is the welfare function $w$ adjusted by the truth$d$ rift $\mu(\theta) / \mu_{0}(\theta)$. For any given posterior belief $\mu$, the likelihood ratio $\mu(\theta) / \mu_{0}(\theta)$ measures the representation of type $\theta$ under $\mu$ relative to its ex ante representation under $\mu_{0}$.

It follows that the Bayes welfare set can be characterized by studying the convex hull of the graph of the vector-valued function, $\hat{\mathrm{w}}: \Delta(\Theta) \mapsto \mathbb{R}^{N}$, where for each $i \in\{1, \ldots, N\}, \hat{\mathrm{w}}_{i}(\mu) \equiv \hat{w}\left(\mu, \theta_{i}\right)$. Indeed, we have the following:

Theorem 2.1 [Doval and Smolin 2024, Theorem 1]. The Bayes welfare set W satisfies the following:

$$
\mathrm{W}=\left\{\mathrm{w} \in \mathbb{R}^{N}:\left(\mu_{0}, \mathrm{w}\right) \in \operatorname{co}(\text { graph } \hat{\mathrm{w}})\right\},
$$

where co denotes the convex hull operator.
Theorem 2.1 provides a geometric characterization of the set W : it is the section at the prior of the convex hull of the graph of the truth-adjusted welfare function $\hat{\mathrm{w}}$. We illustrate Theorem 2.1 through an example:

Example 2.2 Online marketplace. An online marketplace wants to design a recommendation algorithm, directing consumers to buy from sellers in the platform. For simplicity, assume sellers may be of one of two equally likely types: low quality $\theta_{1}$, and high quality $\theta_{2}$. Consumers prefer to buy from high quality sellers. Thus, each seller's profit in the marketplace depends on the likelihood $\mu$ the consumer attaches to the seller being of high quality. In particular, we assume the sellers' profits as a function of consumers' beliefs are as follows:

$$
w(\mu, \theta)=\left\{\begin{array}{ll}
0 & \text { if } \mu \in[0,1 / 3) \\
1 / 2 & \text { if } \mu \in[1 / 3,2 / 3) \\
1 & \text { if } \mu \in[2 / 3,1]
\end{array} .\right.
$$

In this example, the set W then represents the set of profit profiles sellers with different qualities can attain in the marketplace under some information structure.

Figure 1 illustrates the convex hull of the graph of $\hat{\mathrm{w}}$ (Figure 1a) and the Bayes welfare set (Figure 1b) for the online marketplace example. For instance, fully revealing or concealing the sellers' quality is always feasible, so that the full and no-disclosure profiles, $\mathrm{w}^{F D}$ and $\mathrm{w}^{N D}$ are feasible. We highlight some properties of the Bayes welfare set:


- Despite the welfare function being symmetric across seller types, the Bayes welfare set is not symmetric because the adjusted-welfare function is not symmetric. By Bayes rule, when consumers are optimistic about the seller's quality being high, it is more likely they are facing a high rather than a low quality seller.
- In particular, the Bayes welfare set lies above the 45° line: the only Bayes welfare profile equalizing seller profits is the no disclosure one, but it is not Pareto efficient. In other words, fairness-measured by welfare parity-may be at odds with Pareto efficiency.
- The Pareto frontier of the Bayes welfare set is given by its north-east boundary. In particular, the flat segment at the top shows the profits of low-quality sellers can be increased without decreasing those of high-quality sellers.
- The points on the decreasing part of the Pareto frontier can only be generated with at least three signals. By contrast, in standard Bayesian persuasion, two signals are enough in the case of two states. Formally, the analogue of the W in Bayesian persuasion has dimension $N$, whereas the W has dimension $2 N-1$.

Because the Bayes welfare set is convex, it can be alternatively described by its supporting hyperplanes. [Doval and Smolin 2024, Theorem 2] shows the frontier of

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Fig. 1: Constructing the Bayes welfare set in Example 2.2; $\mathrm{w}^{F D}$ and $\mathrm{w}^{N D}$ denote the profit profiles under full and no information, respectively.

the Bayes welfare set can be obtained as the solution to series of Bayesian persuasion problems as in [Kamenica and Gentzkow 2011], in which a utilitarian planner takes the role of the information designer. Concretely, consider the supporting hyperplane of the W in direction $\lambda \in \mathbb{R}^{N} \backslash\{0\}$. Then, the Bayes welfare profiles on the boundary of the W in direction $\lambda$ can be obtained by solving the Bayesian persuasion problem of a sender with indirect utility

$$
\hat{v}_{\lambda}(\mu)=\sum_{\theta \in \Theta} \mu(\theta) \frac{\lambda(\theta)}{\mu_{0}(\theta)} w(\mu, \theta) .
$$

We have found this result very useful in computing the Bayes welfare set in applications (see also [Corrao and Dai 2023] for an application to strategic communication).


## 3. BEYOND THE BASIC MODEL: GROUPS AND DATA

Two assumptions are implicit in the analysis so far. First, we assume the variable the unmodeled outside observer cares about is the same variable on which we condition the payoffs. Consider, however, an employer making hiring decisions based on a candidate's ability. If candidates belong to different groups, basing hiring recommendations on ability impacts the welfare of candidates across different groups. Second, we assume the information structure can arbitrarily condition on an individual's payoff-relevant type. However, because regulation may prevent the disclosure of protected characteristics, such as gender or race, considering algorithms that respect these restrictions is natural whenever $\theta$ includes such characteristics.

Formally, we extend the basic model as follows. We now distinguish between three random variables: an individual's group $g \in G$, the state $\omega \in \Omega$, and data $d \in D$. The first is the variable we condition payoffs on; the second is the variable of interest to the outside observer; the third allows us to capture limits on the information provided. We let $\mathbb{P} \in \Delta(G \times \Omega \times D)$ denote the joint distribution over group-state-
data pairs, and in a slight abuse of notation we denote by $\mathbb{P}(\cdot \mid g)$ and $\mathbb{P}(\cdot \mid d)$ the prior distribution conditional on the individual's group and the data realization, respectively. Below, we denote the marginal of $\mathbb{P}$ on $D$ by $\eta_{0} \in \Delta(D)$. In a slight abuse of notation, we define the welfare function as $w: \Delta(\Omega) \times \Omega \times G \mapsto \mathbb{R}$, with its first argument being the (unmodeled) outside observer's belief $\mu$ about the state $\omega, \mu \in \Delta(\Omega)$. The basic model corresponds to the case in which $\Theta=G=\Omega=D$ and $\mathbb{P}(\omega, d \mid g)=\mathbb{1}[g=\omega=d]$.

To capture the limits data imposes on information provision, an information structure is now defined as a tuple $(\pi, S)$, where $\pi: D \mapsto \Delta(S)$. Given the information policy, belief updating about $(g, \omega, d)$, and hence about $\omega$, depends only on the updated belief about $d$. Specifically, letting $\eta_{s}$ denote the updated belief starting from $\eta_{0}$, after observing signal $s \in S$, the updated belief on $(g, \omega, d)$ is given by $\mathbb{P}(g, \omega \mid d) \eta_{s}(d)$. Without loss of generality, we can write the welfare function as $w_{\dagger}(\eta, \omega, g) \equiv w(\mu(\eta), \omega, g)$.

Given an information structure $(\pi, S)$, the welfare of individuals of group $g$ is:

$$
w_{\Pi}(g)=\sum_{\eta \in \operatorname{supp}(\Pi)} \sum_{s \in S: \eta_{s}=\eta} \sum_{(\omega, d)} \mathbb{P}(\omega, d \mid g) \pi(s \mid d) w_{\dagger}(\eta, \omega, g),
$$

and the Bayes welfare set continues to be defined as the set of Bayes welfare profiles.
The characterization of the Bayes welfare set in the basic model extends verbatim to the more general model, once we observe (the analogue of) the truth-adjusted welfare function now takes the form:

$$
\hat{w}_{\dagger}(\eta, g)=\sum_{(\omega, d)} \mathbb{P}(\omega, d \mid g) \frac{\eta(d)}{\eta_{0}(d)} w_{\dagger}(\eta, \omega, g) .
$$

Equation (7) allows us to provide further insight into the adjusted welfare function in the basic model. The likelihood correction is now based on the variable $d$, highlighting that it corresponds to the variable on which information is being provided. In addition, the presence of additional uncertainty requires averaging over $\omega$ and $d$ using weights $\mathbb{P}(\omega, d \mid g)$. Thus we can immediately extend Theorem 2.1 as:

Theorem 3.1 [Doval and Smolin 2021, Theorem 4]. The Bayes welfare set can be calculated as:

$$
\mathrm{W}=\left\{\mathrm{w} \in \mathbb{R}^{|G|}:\left(\eta_{0}, \mathrm{w}\right) \in \operatorname{co}\left(\operatorname{graph} \hat{\mathrm{w}}_{\dagger}\right)\right\} .
$$

We note again that it is the prior on data, $\eta_{0}$, and not on the states which determines the constraint on how much information can be provided about the state of the world, and hence the limits on how much welfare can be generated via information.

Example 3.2 Data Regulation in Hiring. Consider two equally likely groups of workers, labeled $A$ and $B$. Workers can have one of two ability levels $\Omega=\{0,1\}$. In each group, half of the workers are high ability and half are low ability. Suppose these workers face a competitive job market: if the market's perceived likelihood that their ability is 1 equals $\mu \equiv \mu(1)$, they receive wage equal to $\mu$. Equating workers' welfare to their wages, this means that $w(\mu, \omega, g)=\mathbb{E}_{\mu}[\omega]$.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Fig. 2: Bayes welfare set under different data regimes in Example 3.2: The red circle corresponds to the Bayes welfare set under data regimes (i)-(iii); the blue line is the Bayes welfare set in regime (iv).

Rather than assuming a fixed data structure, we compare the Bayes welfare sets in this setting across two data regimes which can be interpreted as different data regulation policies that limit how much information can be revealed about a worker's ability: data reveals ability, but not group (i.e., $D=\Omega$ ), and data reveals both group and ability (i.e., $D=G \times \Omega$ ). Figure 2 illustrates the Bayes welfare set in each of these regimes.

Whereas in the first regime we can provide meaningful information about ability, that the distribution of ability is independent across groups together with the martingale property of beliefs implies that on average the posterior belief about the ability remains the same as under no information. It follows that in this case the Bayes welfare set consists of the no disclosure profile, $\mathrm{W}=\{(1 / 2,1 / 2)\}$.

Consider now the second regime and an information structure that pools low-ability workers from group A with high-ability workers from group B and fully reveals all other workers. We can represent this as an information structure with signals $\{B 0\},\{A 0, B 1\}$, and $\{A 1\}$, and induced posterior expectations of 0, $\frac{1}{2}$, and 1, respectively. Because different groups induce these signals with different probabilities, each group's welfare is given by:


$$
\begin{aligned}
& \mathrm{w}_{A}=\frac{1}{2} \mathbb{E}[\mu \mid\{A 0, B 1\}]+\frac{1}{2} \mathbb{E}[\mu \mid\{A 1\}]=\frac{1}{2} \frac{1}{2}+\frac{1}{2} 1=\frac{3}{4}, \\
& \mathrm{w}_{B}=\frac{1}{2} \mathbb{E}[\mu \mid\{B 0\}]+\frac{1}{2} \mathbb{E}[\mu \mid\{A 0, B 1\}]=\frac{1}{2} 0+\frac{1}{2} \frac{1}{2}=\frac{1}{4} .
\end{aligned}
$$

In fact, this information structure achieves the maximal possible payoff for group A: It never pools workers from group A with the low ability workers of group B, it never pools the high ability workers from group A with workers from group B, and it pools all high ability workers from group B with the workers from group A. As such, the maximal welfare $\mathrm{w}_{A}$ is $\frac{3}{4}$.

We note, however, that the average payoff across groups is the same across all information structures:

$$
\frac{1}{2} \mathrm{w}_{A}+\frac{1}{2} \mathrm{w}_{B}=\frac{1}{2} \mathbb{E}[\mu \mid g=A]+\frac{1}{2} \mathbb{E}[\mu \mid g=B]=\mathbb{E}[\mu]=\frac{1}{2} .
$$

In other words, information merely redistributes welfare across the groups. Consequently, the information structure that maximizes the welfare of group $A$ minimizes that of group $B$.

These observations together with the symmetry of the setting imply that in the fourth regime the Bayes welfare set is given by:

$$
\mathrm{W}=\left\{\left(\mathrm{w}_{A}, \mathrm{w}_{B}\right) \in[1 / 4,3 / 4]^{2}: \mathrm{w}_{A}+\mathrm{w}_{B}=1\right\} .
$$


## 4. FINAL REMARKS

We conclude by describing alternative applications of our framework as well as some directions for further research.


### 4.1 Applications to information design

By interpreting our welfare function as an individual's type-dependent payoff function, the Bayes welfare set is also the object of interest in more standard information design applications. For instance, the types may represent the private information of an informed principal who can commit to an information structure only after observing her type, as in [Perez-Richet 2014] and [Koessler and Skreta 2023]. Similar constraints appear in the studies of information design without commitment, as in [Lipnowski and Ravid 2020], [Drakopoulos et al. 2022], and [Corrao and Dai 2023]. Thus, the Bayes welfare set can be viewed as a unifying concept that underlies the incentive constraints the equilibrium information structure must satisfy. As we show in our first working paper version, [Doval and Smolin 2021], our tools also open the door to the study of new problems in this literature such as communication equilibrium payoffs in Bayesian persuasion with transparent motives and Bayesian persuasion with an ambiguity averse sender. ${ }^{1}$


### 4.2 Further research

We conclude with three (non-exhaustive) suggestions for future research:
It is well-known that various statistical notions of fairness, such as equalized odds and calibration, are incompatible with each other (cf. [Chouldechova 2017; Kleinberg et al. 2016]). Furthermore, this incompatibility remains even when considering relaxations [Pleiss et al. 2017]. Yet, the Bayes welfare set provides another way to visualize the trade-offs among these competing notions. For instance, one could use the Bayes welfare set to understand which group is hurt the most when imposing either calibration or equalized odds. Similarly, one could consider information structures that preserve some form of privacy-e.g., the algorithm recommendations do not reveal information about group membership-and study the Bayes

[^0]welfare profiles consistent with such restrictions (cf. [Gopalan et al. 2021; Strack and Yang 2024]).

Since the seminal work of [Dughmi and Xu 2016], the computer science literature has made incredible progress in algorithmic Bayesian persuasion (see, e.g., [Babichenko and Barman 2016; Arieli and Babichenko 2019; Banerjee et al. 2024]). Most of this work is concerned with the computational aspects of achieving the sender's preferred payoff, whereas our work focuses on the cross-sectional implications of different information structures for which the sender's average payoff may not be a sufficient statistic.

In many applications, considering constraints on the information structures the planner has access to is natural. The model in Section 3 puts limits on how much information can be provided about the payoff-relevant state. Constraints such as those arising from differential privacy are relevant in many applications and understanding how they shape the choice out of the Bayes welfare set is of interest.


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[^0]:    ${ }^{1}$ [Corrao and Dai 2023] fully characterize the set of communication equilibria with transparent motives.


## Citation and provenance

**Authors:** Laura Doval; Alex Smolin

**Canonical citation:** Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.

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