Algorithmic Game Theory

Motivation

A supplier who must build capacity before a selling season would like to know how large demand will be, and the retailer downstream often knows better. Under a simple wholesale-price contract, however, the retailer gains from overstating demand: more capacity means more sales for her, and the cost of idle capacity falls on the supplier. Since the retailer's message costs nothing and binds nobody, a rational supplier ignores it. That such cheap talk (Crawford and Sobel, 1982) cannot convey information in a one-shot wholesale-price relationship is the benchmark result of Özer, Zheng and Chen (2011), yet firms are observed to share forecasts informally.

Avinadav and Shamir propose an explanation based on partial vertical ownership (PVO): retailers often hold passive equity stakes in their suppliers. A retailer that owns a share $\beta$ of the supplier bears a fraction $\beta$ of the losses her misreport inflicts on him. Their Theorem 2 says exactly how large the stake must be for truthful reporting to be self-enforcing, and how this depends on the gap between the two possible demand levels. This mission formalizes that theorem.

Setting

A retailer sources a component from a supplier. The supplier secures capacity $K$ at unit cost $c$; after demand is realized, the retailer orders and pays the wholesale price $w$ per unit and sells at the retail price $r$, with $0<c<w<r$. Unmet demand is lost, and leftover capacity is worthless.

Demand in market state $i\in\{L,H\}$ is $D_i=\mu_i+\varepsilon$ with $0<\mu_L<\mu_H$, where $\varepsilon$ is a zero-mean random variable with a density, distribution function $F$, and lower support end $\underline{\varepsilon}<0$ satisfying $\mu_L+\underline{\varepsilon}\ge0$. The information gap is $\Delta=\mu_H-\mu_L$. Write $F^{-1}(z)=\inf\{x: F(x)\ge z\}$ and define the critical fractiles $z_s=1-c/w$ and $z_r=1-w/r$.

The retailer may commit to an advance order $q^{adv}$ which the supplier must cover. The supplier's and retailer's expected profits are

$$\pi_s(K\mid\mu_i,q^{adv})=\mathbb E\big[w\min(K,\max(\mu_i+\varepsilon,q^{adv}))-cK\big],$$ $$\pi_r(q^{adv},K\mid\mu_i)=\mathbb E\big[(r-w)\min(\mu_i+\varepsilon,K)-w\max(q^{adv}-\mu_i-\varepsilon,0)\big].$$

A retailer holding the share $\beta\in[0,1]$ of the supplier maximizes the accounting payoff $v_r(q^{adv},K\mid\mu_i)=\pi_r(q^{adv},K\mid\mu_i)+\beta\,\pi_s(K\mid\mu_i,q^{adv})$.

Only the retailer observes the state. In the cheap-talk game she places no advance order and announces a state; a supplier who believes the announcement builds the newsvendor capacity $K_L=\mu_L+F^{-1}(z_s)$ or $K_H=\mu_H+F^{-1}(z_s)$. Truthful reporting requires

$$\text{(12)}\quad v_r(0,K_H\mid\mu_H)\ge v_r(0,K_L\mid\mu_H),\qquad\text{(13)}\quad v_r(0,K_L\mid\mu_L)\ge v_r(0,K_H\mid\mu_L).$$

Formalization targets

Goal: Theorem 2

Let

$$\underline{\underline{\beta}}=\frac{\pi_r(0,K_H\mid\mu_L)-\pi_r(0,K_L\mid\mu_L)}{\pi_s(K_L\mid\mu_L,0)-\pi_s(K_H\mid\mu_L,0)},\qquad R(d)=\frac1d\int_{F^{-1}(z_s)}^{d+F^{-1}(z_s)}F(x)\,dx\quad(d>0).$$

The goal asserts: (12) holds for every $\beta\in[0,1]$; for every $\beta\in[0,1]$, (12) and (13) hold if and only if $\beta\ge\underline{\underline{\beta}}$;

$$\underline{\underline{\beta}}=\frac{z_r}{1-z_r}\left(\frac{1-z_s}{R(\Delta)-z_s}-1\right),\qquad R(\Delta)<1;$$

and for every $\beta\in(0,1]$ the equation $R(d)=z_s+\frac{1-z_s}{1+\beta(1/z_r-1)}$ has a unique root $\underline{\underline{\Delta}}>0$, and truthful cheap talk is possible only if $\Delta\ge\underline{\underline{\Delta}}$.

Milestones

In attack order: the closed forms (2) and (3) of the expected profits; condition (12) always holds; (E.1), that (13) is equivalent to $\beta\ge\underline{\underline{\beta}}$ with positive numerator and denominator; (E.2), the closed form of $\underline{\underline{\beta}}$; and the properties of $R$ (strictly increasing on $(0,\infty)$, limits $z_s$ at $0^+$ and $1$ at $\infty$, values in $(z_s,1)$). A companion item states Proposition 1: $\underline{\underline{\beta}}<1$ if and only if the integrated chain earns less with $K_H$ than with $K_L$ in the low state.

Significance

Theorem 2 identifies an ownership channel through which a one-shot wholesale-price relationship can sustain truthful information exchange, in contrast with the impossibility of cheap talk without such links. The threshold $\underline{\underline{\beta}}$ is the ratio of the low-type retailer's operational gain from lying to the supplier's loss from being misled, and part (ii) turns it into a lower bound on the information gap: a sufficiently small gap rules out cheap talk for every stake. Proposition 1 translates feasibility ($\underline{\underline{\beta}}<1$) into a comparison of the integrated chain's profits. The same comparisons underlie the paper's later analysis of signaling with advance orders, which takes cheap talk as its benchmark.

The result is proved in the source, in Appendix E of the authors' manuscript; it has no machine-checked proof. The mission produces a formal statement in which every profit is an expectation over a general noise distribution, so that the closed forms the paper derives become theorems rather than definitions, together with the supporting analysis of the averaged distribution function $R$.

Difficulty

The algebra after the closed forms is short. The work lies in two places. First, the closed forms (2) and (3) are integration-by-parts identities for expectations of truncated linear functions of $\varepsilon$, under a general law rather than a density on a compact interval, including the case of an upper support end at $+\infty$. Second, the properties of $R$: the paper differentiates $R$ and applies L'Hôpital's rule, which presumes more regularity than the standing assumptions give; positivity of the numerator and denominator of $\underline{\underline{\beta}}$, the strict bounds $z_s<R<1$, and the existence and uniqueness of $\underline{\underline{\Delta}}$ all rest on continuity and strict increase of $F$ near $F^{-1}(z_s)$. The case $\beta=0$ in part (ii) is degenerate: the argument of $R^{-1}$ is then $1$, which $R$ never attains.

Formalization scope

The source is the authors' manuscript of 2022-06-06; theorem numbers, equation numbers and pages are the manuscript's. The model is a structure with real fields $c,w,r,\mu_L,\mu_H,\underline{\varepsilon}$ and a measure $P$ on $\mathbb R$ (the law of $\varepsilon$), with $F=$ cdf P. Every theorem assumes the standing hypotheses:

• $P$ is a probability measure, absolutely continuous with respect to Lebesgue measure;
• $\varepsilon$ is integrable with mean zero (integrability is implicit in "zero-mean");
• $\underline{\varepsilon}<0$, $F(\underline{\varepsilon})=0$, and $F$ is strictly increasing on $\{x\ge\underline{\varepsilon}:F(x)<1\}$ (added: the paper uses $F^{-1}$ as an inverse);
• $0<\mu_L<\mu_H$, $\mu_L+\underline{\varepsilon}\ge0$, and $c<w<r$ together with the added $c>0$.

The profits $\pi_s,\pi_r,v_r$ are Bochner integrals against $P$, $\underline{\underline{\beta}}$ is the plain quotient of profit differences (the first expression of (14)), and conditions (12)–(13) are the payoff comparisons themselves. Defining any of these by the paper's closed forms would make the milestones definitional and is ruled out. $R$ is used only at $d>0$; $\underline{\underline{\Delta}}$ is characterized as the unique positive root rather than through a junk inverse, and part (ii) is stated for $\beta\in(0,1]$. The supply-chain profit $\pi_{sc}$ is taken at advance order $0$, as in the proof of Proposition 1. The prior over market states and the signaling and pooling equilibria (Theorems 3–4) are outside this mission.

Mathlib provides distribution functions, Bochner and interval integrals and limits. A complete development needs layer-cake identities for $\mathbb E[\min(K,\cdot)]$ and $\mathbb E[(\cdot)^+]$ in terms of $\int F$, continuity of the CDF of an absolutely continuous law, and the monotonicity and limits of interval averages of a monotone function; these are reusable well beyond this model. Proofs of any milestone, or of general lemmas they need, are welcome.

Selected references

• T. Avinadav and N. Shamir, Partial Vertical Ownership, Capacity Investment, and Information Exchange in a Supply Chain, Management Science, 2023 (authors' manuscript of 2022-06-06). https://doi.org/10.1287/mnsc.2023.4673
• V. P. Crawford and J. Sobel, Strategic Information Transmission, Econometrica 50(6), 1982. https://doi.org/10.2307/1913390
• Ö. Özer, Y. Zheng and K.-Y. Chen, Trust in Forecast Information Sharing, Management Science 57(6), 2011. https://doi.org/10.1287/mnsc.1110.1348

Prize money in rank-order competitions

Tournaments, races, poker finals and sports leagues end with a ranking of the participants and a sum of prize money to be split among them. The split is usually fixed by a published schedule, and different events use very different schedules: some pay almost everything to the winner, some pay many places, some decay geometrically from one place to the next. Dietzenbacher and Kondratev (arXiv:2008.08985, published in Management Science, 2023) treat the choice of schedule axiomatically: they state fairness, solidarity and consistency properties that a prize allocation rule may satisfy and characterize the families of rules that satisfy them. Their paper has three characterization theorems. Theorem 1 (interval rules) is the subject of a companion mission; this mission formalizes Theorem 2, the characterization of single-parametric rules under local consistency.

Axiomatic characterizations of allocation rules through consistency have a long history in fair division: bankruptcy problems (Aumann and Maschler, 1985), taxation (Young, 1987), bargaining (Lensberg, 1987) and cooperative games (Peleg, 1986; Hart and Mas-Colell, 1989); the paper's §1 surveys this literature. The paper is, by its own account, the first to apply this approach to prize allocation in rank-order competitions.

Setting

Fix a countable set $U$ of at least three potential competitors. A competition $(N,\mathcal R,E)$ consists of a finite set $N\subseteq U$, a ranking $\mathcal R:N\to\{1,\dots,|N|\}$ (a bijection; $i$ is ranked above $j$ when $\mathcal R(i)<\mathcal R(j)$) and a prize endowment $E\ge0$. For $S\subseteq N$ the subranking $\mathcal R_S$ lists $S$ in the order of $\mathcal R$. A prize allocation rule $\varphi$ assigns to each competition nonnegative prizes $\varphi_i(N,\mathcal R,E)$, $i\in N$, with $\sum_{i\in N}\varphi_i(N,\mathcal R,E)=E$.

The axioms of Theorem 2 are:

• anonymity: the prize depends only on $|N|$, the position and $E$, not on who the competitors are;
• order preservation: a higher-ranked competitor never receives less than a lower-ranked one;
• winner strict endowment monotonicity: when $E$ increases, nobody receives less, and the winner receives strictly more;
• local consistency: if a block $S$ of consecutively ranked competitors (the condition $|\mathcal R(i)-\mathcal R(j)|\le|S|-1$ for all $i,j\in S$) re-divides the total prize it received, using the rule on the reduced competition $(S,\mathcal R_S,\sum_{j\in S}\varphi_j(N,\mathcal R,E))$, each member gets the same prize as before.

A representation is a continuous, non-decreasing $f:\mathbb R_+\to\mathbb R_+$ with $f(x)\le x$. The single-parametric rule with representation $f$ pays $x$ to the winner, $f(x)$ to the runner-up, $f(f(x))$ to the third, and in general

$$\varphi_i(N,\mathcal R,E)=f^{(\mathcal R(i)-1)}(x),\qquad\text{where } x\ge0 \text{ solves } \sum_{k=1}^{|N|}f^{(k-1)}(x)=E .$$

Equal Division ($f(x)=x$), Winner-Takes-All ($f=0$), Winner-Takes-Surplus ($f(x)=\min\{a,x\}$) and the arithmetic rule $f(x)=\max\{0,x-1\}$ are examples.

Formalization targets

Goal: Theorem 2 (p. 18)

$$\varphi \text{ satisfies anonymity, order preservation, winner strict endowment monotonicity and local consistency} \iff \varphi \text{ is a single-parametric rule.}$$

Both directions are part of the goal.

Milestones, in the order the proof on pp. 31–32 uses them

1. Converse (p. 31): every single-parametric rule satisfies the four axioms.
2. The representation (p. 31): on a two-competitor competition, the allocation path $E\mapsto(\varphi_1,\varphi_2)$ is the graph $\{(x,f(x))\}$ of a well-defined representation $f$.
3. Two-competitor form (p. 31): by anonymity, the rule is the single-parametric rule with representation $f$ on every competition with two competitors.
4. Propagation (pp. 31–32): if the single-parametric rule $\varphi'$ with representation $f$ first exceeds $\varphi$ at position $i$, then $\varphi'_j\ge\varphi_j$ for every $j>i$.
5. Uniqueness (p. 32): a locally consistent rule that follows $f$ on two competitors coincides with $\varphi'$ everywhere.

The paper's Lemma 1 (endowment monotonicity implies endowment continuity, p. 10) is used at the start of the proof; it is formalized in the companion mission on interval rules, and the milestones here take endowment continuity as a hypothesis.

Companions

• Existence and uniqueness of $x$ (p. 18): for every representation $f$, $|N|\ge1$ and $E\ge0$ the equation $\sum_{k=1}^{|N|}f^{(k-1)}(x)=E$ has exactly one solution $x\ge0$.
• Corollary 5 (p. 19): a single-parametric rule satisfies winner-loser strict order preservation iff $f(x)<x$ for $x>0$; strict order preservation iff $0<f(x)<x$ for $x>0$; strict endowment monotonicity iff $f$ is increasing.
• Corollary 6 (p. 20): anonymity, order preservation, endowment additivity and local consistency characterize the geometric rules $\varphi_i=\lambda^{\mathcal R(i)-1}E/\sum_{k=1}^{|N|}\lambda^{k-1}$, $\lambda\in[0,1]$.

Significance

Theorem 2 reduces an infinite family of allocation schedules, one for every number of competitors and every endowment, to one function of one real variable. A designer who accepts the four axioms only has to choose how the prize of one position determines the prize of the next; the schedule for every field size and every endowment then follows. Corollaries 5 and 6 translate further axioms into shape conditions on that function, and identify the geometric schedules used in practice (the paper fits one to a poker final, p. 20) as the scale-invariant case. Local consistency is the property that a league split into consecutive divisions pays every division the same as the whole league would.

The result is proved in the paper; nothing here is open. No part of it has been formalized before: the platform has no model of prize allocation rules or of consistency axioms over variable populations. The mission produces that model (competitions over a universe of competitors, subrankings, the axioms) and a machine-checked proof of the characterization, including the "readily checked" converse that the paper leaves to the reader.

Difficulty

The converse requires that the parameter $x$ be well defined: the map $x\mapsto\sum_{k}f^{(k-1)}(x)$ must be shown continuous, strictly increasing and onto $\mathbb R_+$, so that it can be inverted, and local consistency then requires identifying the parameter of a reduced competition with an iterate of the original one. In the forward direction the representation $f$ is defined only implicitly, as the second coordinate of the two-competitor allocation path; its continuity and monotonicity come from inverting the strictly increasing winner's prize $E\mapsto\varphi_1(N,\mathcal R,E)$, which needs both endowment continuity and the strict part of winner strict endowment monotonicity. Without the strict part the runner-up's prize is not a function of the winner's, and the paper's Example 5 shows that local consistency alone then does not determine the rule from its two-competitor restriction. The extension from two to many competitors is a comparison argument over positions that uses local consistency only on adjacent pairs.

Formalization scope

• Competitions. A ranking is a duplicate-free nonempty List U, best competitor first; $N$ is its set of entries and $\mathcal R(i)$ is the 1-based position pos l i. The subranking $\mathcal R_S$ is the sublist of entries in $S$. $U$ is any type with decidable equality; the goal and the corollaries carry the page's standing assumptions [Countable U] and the existence of three distinct elements. Milestones that do not need them omit them.
• Rules are functions List U → ℝ → U → ℝ. A prize allocation rule is nonnegative and exhausts $E$ on every ranking and every $E\ge0$; values elsewhere are irrelevant, and every axiom quantifies only over rankings, $E\ge0$ and competitors in $N$.
• Representations are functions on $\mathbb R$ constrained on $[0,\infty)$ only: ContinuousOn and MonotoneOn on Set.Ici 0, and $0\le f(x)\le x$ for $x\ge0$. Requiring global continuity or monotonicity on $\mathbb R$ would shrink the family and is not done. The sum $\sum_{k=1}^{|N|}f^{(k-1)}(x)$ uses the iterate f^[k] with the index shifted to $k=0,\dots,|N|-1$; "$x$ is chosen such that" is an existential, and uniqueness of $x$ is a separate item.
• Trivializing encodings ruled out. The rule is defined on every competition over $U$, not on a fixed competitor set; local consistency ranges over exactly the blocks of consecutive positions (using full consistency instead would make the "if" direction false for most $f$); the goal is a biconditional; the family requires continuity of $f$ and is not restricted to a closed form.
• Infrastructure. The model and the axioms shared with the companion missions on interval and parametric rules are one common set of definitions; this mission adds three: local consistency, single-parametric rules, and geometric rules. Lemma 1 and the facts about inverting a continuous strictly increasing function on $\mathbb R_+$ are reusable beyond this mission. Proofs of any item, and of the companions, are welcome.

Selected references

• B. J. Dietzenbacher and A. Y. Kondratev, Fair and Consistent Prize Allocation in Competitions, arXiv:2008.08985v2, 2021; Management Science, 2023. https://arxiv.org/abs/2008.08985 , https://doi.org/10.1287/mnsc.2022.4520
• R. J. Aumann and M. Maschler, Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36, 1985. https://doi.org/10.1016/0022-0531(85)90102-4

Motivation

Platform markets often feature a Superstar: a single complement whose presence moves consumers, such as a blockbuster game on a console or a star streamer on a video platform. Whether such a Superstar signs an exclusive deal with one platform or stays available on both is a recurring question for competition authorities and for platform managers. Empirical work documents both sides of the effect. Exclusive titles help small consoles challenge incumbents (Lee 2013). Superstar apps generate demand-discovery spillovers for small developers (Ershov 2020, NET Institute working paper). When a star streamer switched platforms, viewers followed and other content producers repositioned (Förderer & Gutt 2021, working paper).

Carroni, Madio & Shekhar (Management Science, 2024) build a two-sided Hotelling model with three groups of agents: consumers, a fringe of small complementors, and the Superstar. The model asks how Superstar exclusivity changes prices, participation and welfare. This mission formalizes the paper's analysis of the complementor side under vertical separation. The analysis runs through the equilibrium prices (Lemmas 1–3) and the sorting of complementors into multihomers, singlehomers and inactive firms (Proposition 1). It ends in the welfare result that exclusivity raises complementors' total surplus (Proposition 3).

Setting

Two platforms $1$ and $2$ sit at the ends of a Hotelling line. A unit mass of consumers has types $m\in[-\bar m,\bar m]$, distributed with cumulative distribution function $F$ and density $f$. The type $m$ measures the preference for platform $2$ over platform $1$. The Superstar is worth $\phi$ to a consumer of a platform that hosts it, and $g_i\in\{0,1\}$ records its presence on platform $i$. A consumer of type $m$ joins platform $1$ rather than $2$ when

$$m\le \tilde m=\phi(g_1-g_2)-(p_1-p_2)+\theta(N_1-N_2),$$

where $p_i$ are the consumer prices, $N_i$ the masses of complementors on the platforms, and $\theta>0$ the value of a unit mass of complementors to a consumer. Platform $1$'s consumer share is $D=F(\tilde m)$ and platform $2$'s is $1-D$.

Complementors have an opportunity cost $k\ge0$ of joining each platform, distributed with mass function $\Lambda$ and density $\lambda$. A complementor receives $\gamma>0$ per unit mass of consumers, so it joins platform $i$ if and only if $k\le\gamma D_i$, and $N_i=\Lambda(\gamma D_i)$. A complementor that joins both platforms pays $k$ twice.

Expectations are fulfilled: the share $D$ solves the fixed-point equation

$$D=F\bigl(\phi(g_1-g_2)-(p_1-p_2)+\theta[\Lambda(\gamma D)-\Lambda(\gamma(1-D))]\bigr).$$

Platforms choose prices simultaneously to maximise gross profit $p_iD_i$. Under exclusivity the Superstar is on platform $1$ only ($g=(1,0)$); under non-exclusivity it is on both ($g=(1,1)$). The paper's standing assumptions are smoothness and symmetry of $F$, a positive nondecreasing density $\lambda$, and Assumption 3: $f\gamma\theta[\lambda(\gamma F)+\lambda(\gamma(1-F))]<1$, which keeps cross-group externalities below the differentiation parameter. The complementors' total surplus at shares $(D_1,D_2)$ is

$$FS(D_1,D_2)=\int_0^{\gamma D_1}(\gamma D_1-k)\lambda(k)\,dk+\int_0^{\gamma D_2}(\gamma D_2-k)\lambda(k)\,dk .$$

Formalization targets

Goal: Proposition 3

For $\phi>0$, let $D(\phi)$ be platform $1$'s share in the selected exclusivity equilibrium and $E$ the share in the symmetric non-exclusive equilibrium. Then

$$FS\bigl(E,1-E\bigr)<FS\bigl(D(\phi),1-D(\phi)\bigr).$$

The statement fixes no functional forms and no constants. It holds for every model satisfying the assumptions and every $\phi>0$.

Milestones

1. Lemma 1. At every interior equilibrium, $p_1=F(m^\star)\{1/f(m^\star)-\gamma\theta[\lambda(\gamma D_1)+\lambda(\gamma D_2)]\}$, and $p_2$ is the same expression with $1-F(m^\star)$ in place of $F(m^\star)$.
2. Lemma 2. Under non-exclusivity at equal prices, $D=\tfrac12$, $m^\star=0$ and $p=1/(2f(0))-\gamma\theta\lambda(\gamma/2)$. All active complementors multihome.
3. Lemma 3 (demand and complementor part). Under exclusivity, $D^\star_1(1,0)>D^\star_2(0,1)$ and $\Lambda(\gamma D^\star_1)>\Lambda(\gamma D^\star_2)$.
4. Proposition 1. Under exclusivity, complementors with $k\in(0,\gamma D^\star_2]$ multihome, those with $k\in(\gamma D^\star_2,\gamma D^\star_1]$ singlehome on platform $1$, and the rest are inactive. More complementors are active than under non-exclusivity.
5. Decomposition of $\Delta FS$ (Section 4.3) into the contributions of multihomers, of switchers from multihoming to singlehoming, and of entrants.

Significance

The result says that a Superstar's exclusive contract, usually discussed as a tool of foreclosure against the rival platform, benefits the fringe of complementors taken as a whole. Some complementors stop paying for a second platform. Others, inactive without exclusivity, enter on the favoured platform. The comparison feeds into the paper's welfare analysis (consumer surplus, Proposition 4) and into its account of when the Superstar chooses exclusivity (Proposition 5). It supplies a theoretical counterpart to the empirical spillover findings cited above.

The paper's proofs are informal in places that matter. Equilibrium selection under fulfilled expectations is described in words only, and the derivative argument for Lemma 3 tracks an equilibrium as a function of $\phi$ without saying which one. A formal development has to fix these choices, and this mission fixes them explicitly (see Formalization scope). None of these results has been machine-checked before. The fixed-point and first-order machinery for two-sided fulfilled-expectations demand would also serve other platform-competition models in the Armstrong (2006) tradition.

Difficulty

The surplus comparison itself is elementary once the equilibrium shares are known. The work lies in the equilibrium shares. Fulfilled-expectations demand is defined by a fixed point that can have several solutions, so neither the share at a deviation price nor the equilibrium share is a priori a function of prices. The obvious first step, "impose symmetry", does not prove that the non-exclusive share is $\tfrac12$. One must show that $\tfrac12$ is the only interior fixed point at equal prices. Likewise, the exclusive share exceeds $\tfrac12$ only for the equilibrium reached continuously from the symmetric one. Coordination on the disfavoured platform is a fulfilled-expectations outcome the model does not exclude. Lemma 1 needs a differentiable local branch of the fixed point at a deviation, which is where Assumption 3 enters.

Formalization scope

Everything is real-valued. The model is a structure with $\bar m,\gamma,\theta>0$, functions $F,f,\Lambda,\lambda:\mathbb R\to\mathbb R$, and the following hypotheses:

• $F$ is continuous and monotone, $0$ left of $-\bar m$ and $1$ right of $\bar m$, $C^2$ with $F'=f>0$ on $(-\bar m,\bar m)$, and symmetric, $F(-x)=1-F(x)$;
• $\Lambda(0)=0$, $\Lambda$ is $C^2$ with $\Lambda'=\lambda$, and $\lambda>0$ is nondecreasing on $[0,\gamma]$;
• Assumption 3 holds on $(-\bar m,\bar m)$.

Several conventions depart from the paper's wording:

• The paper asks for $\lambda>0$, $\lambda'\ge0$ on all of $[0,\infty)$ for a distribution function $\Lambda$, which no distribution satisfies. These conditions are imposed on $[0,\gamma]$, the only range on which $\Lambda$ is evaluated.
• The bounds on $f'$ in Assumption 2 are dropped, which only strengthens the statements.
• "$\bar m$ large enough" is replaced by interiority ($0<D<1$) of every equilibrium considered.
• The standalone value $v$ cancels and is omitted.

Fulfilled-expectations demand is a relation, not a chosen function. A price equilibrium is a Nash equilibrium in prices in which no deviation is profitable for any fulfilled-expectations share it induces. Lemma 1's formulas are a theorem about this equilibrium, not its definition. The symmetric non-exclusive outcome is the interior equilibrium at equal prices. The exclusive outcome is the end point, at value $\phi$, of a continuous path of interior equilibria of the games with Superstar value $t\in[0,\phi]$, starting from equal prices at $t=0$. Platform $2$'s share is always $1-D$.

The goal must not be posed with $D(\phi)>\tfrac12$, $D(\phi)\ne\tfrac12$ or $E=\tfrac12$ as a hypothesis: since $D\mapsto FS(D,1-D)$ is strictly convex and symmetric about $\tfrac12$, any of these would reduce it to a calculus exercise. The price ranking in Lemma 3 is not part of the mission. Its proof rests on bounds on $f'$ that the paper states only as functions of unspecified equilibrium arguments.

The needed infrastructure is the implicit function theorem in one variable, interval integrals of continuous functions, and the intermediate value theorem. Contributions are welcome on any milestone, on alternative proofs of the goal, and on reusable lemmas about uniqueness of fulfilled-expectations fixed points.

Selected references

• Carroni, E., Madio, L., Shekhar, S., Superstar Exclusivity in Two-Sided Markets, Management Science 70(2):991–1011, 2024; accepted manuscript at IRIS Bologna. https://doi.org/10.1287/mnsc.2023.4720, http://hdl.handle.net/11585/922035
• Armstrong, M., Competition in Two-Sided Markets, RAND Journal of Economics 37(3):668–691, 2006. https://doi.org/10.1111/j.1756-2171.2006.tb00037.x
• Fudenberg, D., Tirole, J., Customer Poaching and Brand Switching, RAND Journal of Economics 31(4):634–657, 2000. https://doi.org/10.2307/2696352
• Lee, R. S., Vertical Integration and Exclusivity in Platform and Two-Sided Markets, American Economic Review 103(7):2960–3000, 2013. https://doi.org/10.1257/aer.103.7.2960
• Ershov, D., Competing with Superstars in the Mobile App Market, NET Institute Working Paper, 2020.
• Förderer, J., Gutt, D., The Effects of Platform Superstars on Content Production: Evidence from Ninja, working paper, 2021.

Motivation

Bayesian persuasion studies a sender who observes a state of the world and commits in advance to a randomized disclosure rule, in order to influence the action of a receiver who updates her belief by Bayes' rule. Since Kamenica and Gentzkow (2011) the model has been applied to platform design, recommendation systems, queue information and rating systems. Almost all of this theory assumes the receiver is an expected-utility maximizer, so that her payoff from each action is linear in her belief.

Many receivers are not. A customer who joins a queue only if the mean of her waiting time plus a multiple of its standard deviation is below a threshold has a utility that is nonlinear in her belief. Anunrojwong, Iyer and Lingenbrink (2023) call such receivers risk-conscious and show that, for them, the standard revelation-principle reduction to action recommendations fails. Their first main result replaces it with a convex program over the joint distribution of state and action. This mission formalizes that result and the lemmas leading to it.

Setting

Let $\Omega$ be a finite set of states and $\Delta(\Omega) \subseteq \mathbb{R}^{\Omega}$ the simplex of beliefs, with the Euclidean topology. Sender and receiver share a prior $\mu^* \in \Delta(\Omega)$. The receiver chooses an action from a finite nonempty set $A$. Her utility is an arbitrary function $\rho : \Delta(\Omega) \times A \to \mathbb{R}$; the sender's utility is $v : \Omega \times A \to \mathbb{R}$.

For each action $a$, the beliefs at which $a$ is optimal for the receiver form the set

$$P_a = \{\mu \in \Delta(\Omega) : a \in \arg\max_{a' \in A} \rho(\mu, a')\}.$$

The paper assumes throughout (Assumption 1) that every $P_a$ is closed.

A signaling scheme consists of a measurable signal space $S$ and a probability measure $\pi$ on $\Omega \times S$ whose $\Omega$-marginal is $\mu^*$. After signal $s$ the receiver holds the posterior $\mu_s \in \Delta(\Omega)$ given by Bayes' rule, and plays an optimal strategy $a(\cdot)$, with $a(s) \in \arg\max_a \rho(\mu_s, a)$, that is $\mu_s \in P_{a(s)}$. The sender's persuasion problem (3) is to maximize $\mathbb{E}_\pi[v(\bar\omega, a(\bar s))]$ over signal spaces, schemes and optimal receiver strategies; when the receiver is indifferent, the sender picks the strategy she prefers.

A probability measure $\eta$ on $\Delta(\Omega)$ is Bayes-plausible if its mean is the prior: $\mathbb{E}_\eta[\bar s] = \mu^*$.

Formalization targets

Goal: Theorem 1

The sender's problem (3) can be optimized by solving the convex program

$$\max_{\{t_a : a \in A\}} \sum_{\omega \in \Omega} \sum_{a \in A} t_a(\omega)\, v(\omega,a) \quad \text{s.t.} \quad \sum_{a \in A} t_a(\omega) = \mu^*(\omega)\ (\omega \in \Omega), \qquad t_a \in \operatorname{Conv}(P_a \cup \{0\})\ (a \in A). \qquad (7)$$

It is formalized as an equality of sets: the values achievable in (3) are exactly the objective values of feasible points of (7). This says that every scheme yields a feasible point of (7) with the same value, and every feasible point is realized by a scheme. In particular the supremum of (3) equals the maximum of (7), and an optimal solution of (7) gives an optimal scheme.

Milestones

1. Lemma 1 (Aumann–Maschler, Kamenica–Gentzkow): schemes whose signals are beliefs and satisfy $\mu_s = s$ correspond to Bayes-plausible measures.
2. Lemma 5: the belief-recommendation problem (4) is equivalent to problem (14) over Bayes-plausible measures.
3. Lemma 2: a Bayes-plausible measure and an optimal strategy give action probabilities $b_a \in [0,1]$ and mean-posteriors $m_a \in \operatorname{Conv}(P_a)$ with $\sum_a b_a m_a = \mu^*$ and the same sender value; the converse also holds.

Further result: Theorem 2

Let $\Upsilon_a$ be the states at which $a$ is sender-optimal, assume these sets partition $\Omega$, and set $t^*_a = \mu^* \mathbb{1}_{\Upsilon_a}$. Then the sender can fully persuade the receiver (almost surely, the receiver takes a sender-optimal action) if and only if $t^*$ is feasible for (7), i.e. $t^*_a \in \operatorname{Conv}(P_a \cup \{0\})$ for all $a$.

Significance

The result. Problem (3) optimizes over arbitrary signal spaces and joint distributions, so it is not finite-dimensional. Program (7) has $|A|$ vector variables in $\mathbb{R}^{\Omega}$, each in a convex set that depends only on the receiver's utility. Its variable $t_a(\omega)$ is the joint probability of state $\omega$ and action $a$. The program gives the paper's bound on the number of signals needed (Proposition 1), its characterization of when persuasion helps (Proposition 2), the full-persuasion criterion (Theorem 2) and, after replacing $\operatorname{Conv}(P_a \cup \{0\})$ by a polytope, the linear program for binary persuasion (Theorem 3, the companion mission). For expected-utility receivers each $P_a$ is convex, and (7) reduces to the familiar obedience-constrained linear program.

Formalizing it. The results are proved in the paper; none is formalized. A formal development fixes what "can be optimized by solving" means as a set equality and handles general measurable signal spaces with posteriors defined only almost everywhere. It also exposes two places where the printed lemmas need correction (see the scope section).

Difficulty

The usual revelation-principle shortcut would merge all signals that lead to the same action. For a risk-conscious receiver this can change the response: averaging posteriors in $P_a$ can produce a belief outside $P_a$, since $P_a$ need not be convex. The central geometric difficulty is relating a distribution of posteriors supported on a closed, potentially nonconvex $P_a$ to a finite-dimensional variable in $\operatorname{Conv}(P_a \cup \{0\})$. The converse must preserve both Bayes plausibility and receiver optimality. With general signal spaces, the posterior is determined only almost everywhere, while the stated strategy condition applies at every signal.

Formalization scope

• Representation. $\Omega$ and $A$ are Fintypes, $A$ nonempty, and $\Omega$ carries a measurable space in which singletons are measurable. Beliefs are functions $\Omega \to \mathbb{R}$ in stdSimplex ℝ Ω, Belief Ω is the simplex as a subtype, $\operatorname{Conv}$ is convexHull ℝ, and $0$ is the zero function. $\rho$ is a function on $\mathbb{R}^{\Omega}$, used only on the simplex.
• Signal spaces in (3) range over all measurable types. A scheme is a probability measure on $\Omega \times S$ with $\pi(\{\omega\} \times S) = \mu^*(\omega)$.
• Posteriors are any version of Bayes' rule: simplex-valued for every signal, measurable, and satisfying $\pi(\{\omega\} \times B) = \int_B \mu_s(\omega)\,\pi_S(ds)$. Quotients $\pi(\omega,s)/\sum_{\omega'} \pi(\omega',s)$ with Lean's $x/0 = 0$ are not used. "$\mu_s = s$" (Lemmas 1 and 5) means the identity is such a version, an almost-sure statement; the paper's "for all $s$" in Lemma 1's converse is proved only almost surely.
• Strategies are measurable (each event $\{a(\bar s) = b\}$ is measurable) and optimal at every signal. The sender's value is not computed with a fixed tie-breaking rule, so sender-preferred selection is kept.
• Assumption 1 is a hypothesis of every statement that involves $\rho$.
• Corrections to Lemma 2. If some $P_a$ is empty, $\operatorname{Conv}(P_a)$ is empty and the printed forward direction fails, so $m_a$ is left free there; then $b_a = 0$ necessarily. The printed converse fails for pure strategies when two actions must share a posterior, so it is stated for optimal mixed strategies, which the paper allows on p. 11.
• Ruled out: quantifying (3) only over belief signals with $\mu_s = s$ would turn Theorem 1 into Lemmas 5 and 2 and drop the coalescence step. The goal therefore quantifies over arbitrary signal spaces. Stating only "the optimal values are equal" through a real supremum would also be weaker, and is not done.
• Infrastructure needed: means of probability measures on compact sets in $\mathbb{R}^n$ lying in the convex hull (Convex.integral_mem, compactness of convex hulls of compact sets), Carathéodory's theorem (convexHull_eq_union), finite discrete measures on $\Omega \times S$, and conditional densities with respect to the signal marginal. The Bayes-plausibility and splitting lemmas are reusable in any persuasion development. Proofs of the milestones, of the goal, and of Theorem 2 are all welcome.

Selected references

• J. Anunrojwong, K. Iyer, D. Lingenbrink, Persuading Risk-Conscious Agents: A Geometric Approach, arXiv:2208.03758v3, 2023; Operations Research, 2024. https://arxiv.org/abs/2208.03758v3
• E. Kamenica, M. Gentzkow, Bayesian Persuasion, American Economic Review 101(6), 2011. https://doi.org/10.1257/aer.101.6.2590
• D. Bergemann, S. Morris, Bayes Correlated Equilibrium and the Comparison of Information Structures in Games, Theoretical Economics 11(2), 2016. https://doi.org/10.3982/TE1808

Motivation

In Bayesian persuasion a sender who observes the state of the world commits to a signaling scheme, a receiver updates her belief by Bayes' rule and acts, and the sender chooses the scheme that maximizes her own expected payoff (Kamenica and Gentzkow 2011). The classical theory assumes the receiver maximizes expected utility, so her payoff is linear in her belief. Many applications violate this: a customer who joins a queue only if the mean of her waiting time plus a multiple of its standard deviation is below a threshold, a commuter who values route reliability, a buyer who dislikes uncertainty about a product. Anunrojwong, Iyer and Lingenbrink (arXiv:2208.03758v3) call such receivers risk-conscious: the receiver's utility $\rho(\mu,a)$ of action $a$ at belief $\mu$ is an arbitrary function of $\mu$.

The paper's first result reduces the sender's problem, an optimization over distributions of posteriors, to a finite-dimensional convex program (program (7)). This mission concerns its second main result: in the practically important binary persuasion setting (two actions, the sender always weakly prefers action $1$), when the region of beliefs at which the receiver rejects action $1$ is convex, program (7) collapses further to a linear program whose feasible region is the convex hull of finitely many explicit beliefs.

Setting

Let $\Omega$ be a finite set of states and $\Delta(\Omega)\subseteq\mathbb R^{\Omega}$ the simplex of beliefs, with the Euclidean topology. For $\omega\in\Omega$, $e_\omega$ is the point-mass belief at $\omega$; states are identified with their point masses. The prior is $\mu^*\in\Delta(\Omega)$. The receiver has actions $A=\{0,1\}$ and utility $\rho(\mu,a)$. The optimality regions are

$$P_a=\{\mu\in\Delta(\Omega):\ a\in\arg\max_{a'\in A}\rho(\mu,a')\},$$

and the differential utility is $\hat\rho(\mu)=\rho(\mu,1)-\rho(\mu,0)$, so $P_1=\{\hat\rho\ge 0\}$ and $P_0=\{\hat\rho\le 0\}$ inside $\Delta(\Omega)$. The sender's utility is $v(\omega,a)$ with $v(\omega,1)\ge v(\omega,0)$ for every $\omega$.

Two assumptions are used.

• Assumption 1 (standing throughout the paper): $P_0$ and $P_1$ are closed.
• Assumption 2: the non-adoption region $P_1^c=\{\mu\in\Delta(\Omega):\hat\rho(\mu)<0\}$ is convex.

From these data the paper builds finitely many beliefs. $K_1=\{\omega: e_\omega\in P_1\}$ and $K_0=\{\omega:e_\omega\in P_0\}$ are the states where action $1$, respectively $0$, is optimal under full information, and $L_0=K_0\setminus K_1$ the states where action $0$ is uniquely optimal. For $\omega_0\in L_0$ and $\omega_1\in K_1$,

$$\gamma(\omega_0,\omega_1)=\sup\{g\in[0,1]:\ g e_{\omega_0}+(1-g)e_{\omega_1}\in P_1\},\qquad \chi(\omega_0,\omega_1)=\gamma(\omega_0,\omega_1)e_{\omega_0}+(1-\gamma(\omega_0,\omega_1))e_{\omega_1},$$

the mixture of $\omega_0$ and $\omega_1$ with the most weight on $\omega_0$ at which action $1$ is still optimal, and $K_{01}=\{\chi(\omega_0,\omega_1):\omega_0\in L_0,\ \omega_1\in K_1\}$.

A candidate solution is a pair $t=(t_0,t_1)$ of vectors in $\mathbb R^{\Omega}$, with $t_a(\omega)$ the joint probability of state $\omega$ and action $a$. The sender's objective is $\sum_\omega v(\omega,1)t_1(\omega)+v(\omega,0)t_0(\omega)$. Program (7) maximizes it subject to $t_0+t_1=\mu^*$ and $t_a\in\operatorname{Conv}(P_a\cup\{0\})$; program (10) subject to $t_0+t_1=\mu^*$, $t_1\in\operatorname{Conv}(K_1\cup K_{01}\cup\{0\})$ and $t_0\in\operatorname{Conv}(L_0\cup\{0\})$.

Formalization targets

Goal: Theorem 3

Under Assumptions 1 and 2, with $v(\omega,1)\ge v(\omega,0)$ for all $\omega$:

$$\text{(i)}\ \ t \text{ feasible for (10)} \Rightarrow t \text{ feasible for (7)};\qquad \text{(ii)}\ \ t \text{ feasible for (7)} \Rightarrow \exists\, t' \text{ feasible for (10)},\ \ \text{obj}(t)\le\text{obj}(t').$$

Together these say that (7) and (10) have the same optimal value and every optimal solution of (10) is optimal for (7): (7) "can be optimized by solving" (10).

Milestones

1. $\chi(\omega_0,\omega_1)\in P_0\cap P_1$ for $\omega_0\in L_0$, $\omega_1\in K_1$ (Section 5.1, under Assumption 1).
2. Lemma 6: $\Delta(\Omega)=\operatorname{Conv}(L_0\cup K_{01})\cup\operatorname{Conv}(K_1\cup K_{01})$, with no assumption on $\rho$.
3. Lemma 7: under Assumption 2, $\operatorname{Conv}(P_1)=\operatorname{Conv}(K_1\cup K_{01})$.

Further results on the same objects

• Lemma 3: if $\hat\rho$ is quasiconvex on $\Delta(\Omega)$, Assumption 2 holds.
• Proposition 3: if $v(\omega,1)-v(\omega,0)=\bar v>0$ for all $\omega$ and a strict total order $\prec$ satisfies the monotonicity Assumption 3, every optimal solution of (10) has a threshold $\omega_0\in L_0$ with $t_1(\omega)=\mu^*(\omega)$ for $\omega\prec\omega_0$ and $t_1(\omega)=0$ for $\omega_0\prec\omega$.
• With strict preference $v(\omega,1)>v(\omega,0)$ for all $\omega$, every optimal solution of (7) is feasible for (10).

Significance

Theorem 3 replaces an optimization over a convex set described only implicitly through $\rho$ by a linear program over an explicit polytope. It identifies a canonical signal set: irrespective of the prior, an optimal scheme only needs to induce beliefs in $\Omega\cup K_{01}$, that is, either reveal the state or leave the receiver uncertain between one state of $L_0$ and one of $K_1$. Lemma 7 gives a simple full-persuasion test, $\mu^*\in\operatorname{Conv}(K_1\cup K_{01})$, and Proposition 3 a threshold structure of the induced actions that the paper uses in its queueing application. For expected-utility receivers the corresponding statements reduce to classical facts about concavification; the content here is that they survive non-linear utilities under a single convexity condition.

On the formal side, the results are proved in the paper and nothing about them is machine-checked. A complete development would give a verified reduction of a family of non-linear persuasion problems to linear programs, and a reusable library of convex-hull facts on the simplex (decompositions of the simplex into hulls of point masses and segment points). The formalization also corrects the paper at one point: the proof of Theorem 3 asserts that every optimal solution of (7) is feasible for (10), which fails when the sender is indifferent between actions in some state; the goal is stated in the form the proof establishes.

Difficulty

The inclusion of (10)'s feasible region in (7)'s is immediate once $K_{01}\subseteq P_1$. The content lies in the converse: $P_1$ is an arbitrary closed set whose complement in the simplex is convex, and one must show that every point of $P_1$ is a convex combination of point masses in $K_1$ and the boundary points $\chi(\omega_0,\omega_1)$ on the edges of the simplex. The natural first idea, to argue with extreme points of $\operatorname{Conv}(P_1)$, does not work directly: $P_1$ need not be convex, its convex hull need not be closed a priori, and the extreme points of $P_1$ are not in general on edges of the simplex. Assumption 2 constrains only the complement of $P_1$, so it has to be transported to a statement about $P_1$ itself. A second difficulty is that $t_0$ in a solution of (7) may put weight on states in $K_0\cap K_1$ or on interior beliefs, and it must be moved to $t_1$ without lowering the objective.

Formalization scope

• Beliefs are functions Ω → ℝ with [Fintype Ω] [DecidableEq Ω]; $\Delta(\Omega)$ is Mathlib's stdSimplex ℝ Ω; $e_\omega$ is Pi.single ω 1; actions are Fin 2; $\operatorname{Conv}$ is convexHull ℝ; $0$ is the zero vector.
• The receiver's utility is ρ : (Ω → ℝ) → Fin 2 → ℝ, and $P_a$ is defined from it by Eq. (1), intersected with the simplex; values of ρ off the simplex are irrelevant. Assumption 1 is IsClosed (P ρ a) for both actions.
• $K_0,K_1,L_0$ are sets of states; they enter convex hulls through their images under $e$. $\gamma$ is a real sSup, used only for $\omega_0\in L_0$, $\omega_1\in K_1$, where the set contains $0$ and is bounded by $1$.
• Programs (7) and (10) are feasibility predicates on t : Fin 2 → Ω → ℝ with a common objective. Theorem 3 is stated as (i) and (ii) above, never through a supremum of objective values. A formalization in which (10) only needs to have some feasible point, or in which the supremum of an empty or unbounded set of values is compared, would be trivial or meaningless and is not the target.
• Proposition 3 takes the order of Assumption 3 as a relation given with the data, and assumes $L_0\neq\emptyset$, which the paper uses implicitly.
• Needed infrastructure: convex hulls of finite sets of point masses in the simplex, Carathéodory-type finite decompositions (available in Mathlib), and the decomposition of a convex combination along a segment. Proofs of the milestones, alternative proofs of Lemma 7 (for example by a separation argument), and proofs of the further results are all welcome.

Selected references

• Anunrojwong, J., Iyer, K., Lingenbrink, D., Persuading Risk-Conscious Agents: A Geometric Approach, arXiv:2208.03758v3, 2023 (journal version: Operations Research, 2024). https://arxiv.org/abs/2208.03758v3
• Kamenica, E., Gentzkow, M., Bayesian Persuasion, American Economic Review 101(6), 2011. https://doi.org/10.1257/aer.101.6.2590

Motivation

Ride-hailing platforms such as Uber and Lyft match riders with self-scheduling drivers who decide for themselves whether to work. Two features make capacity planning on such a platform different from classical fleet management. Demand is spatially imbalanced: more trips start in some locations than end there, so cars accumulate where they are not needed unless they drive back empty. And capacity is endogenous: the number of participating drivers depends on what drivers expect to earn, which in turn depends on how the platform uses them.

Afèche, Liu and Maglaras (MSOM 2023) study how much a platform gains from two operational controls under these conditions: admission control (rejecting some ride requests, depending on destination) and centralized repositioning (directing idle drivers to move empty between locations). The benchmark of their comparison is the regime in which the platform has both controls, called centralized control (regime $C$). This mission formalizes the paper's characterization of that benchmark: how a fixed number of drivers should be allocated, what each driver then earns, and why the resulting driver equilibrium is the platform's optimum.

Setting

There are two locations and four routes $lk$, $l,k\in\{1,2\}$. Route $lk$ has potential demand rate $\Lambda_{lk}$ and travel time $t_{lk}>0$. The price is $p$ per unit of travel time; the platform keeps a share $\gamma\in(0,1)$ and drivers keep $\bar\gamma p$, with $\bar\gamma=1-\gamma$, while paying a driving cost rate $c$, with $\bar\gamma p-c>0$. Demand is imbalanced: $0<\Lambda_{12}<\Lambda_{21}$ (Assumption 1). Repositioning is profitable: $c\,t_{12}<t_{21}(\bar\gamma p-c)$ and $c\,t_{21}<t_{12}(\bar\gamma p-c)$ (Assumption 2).

A pool of $N$ potential drivers has opportunity cost rates drawn from a continuous distribution with CDF $F$, strictly increasing on $[0,\bar c_o]$ with $\bar c_o\ge p-c$, $F(0)=0$ and $F(\bar c_o)=1$ (Assumption 3). If drivers expect a profit rate $\pi$, then $n=NF(\pi)$ of them participate.

In steady state the platform splits a driver capacity $n$ into service capacity $s_{lk}\le S_{lk}:=\Lambda_{lk}t_{lk}$ on each route, repositioning capacity $r_{12},r_{21}$ (empty driving) and queueing capacity $q_1,q_2$ (idle drivers). With totals $\bar s$, $\bar r$, $\bar q$ and $\bar S=\sum_{lk}S_{lk}$, the set $\mathcal C_C(n)$ of feasible allocations is

$$\frac{s_{12}+r_{12}}{t_{12}}=\frac{s_{21}+r_{21}}{t_{21}},\qquad \bar s+\bar r+\bar q=n,\qquad 0\le s\le S,\ r\ge0,\ q\ge0 .$$

The first equation balances the flows of cars between the locations. The platform earns $\Pi(s)=\gamma p\bar s$; each driver earns

$$\pi(s,r,n)=\frac{(\bar\gamma p-c)\bar s-c\bar r}{n}.$$

The platform's problem (13) maximizes $\Pi(s)$ over $(s,r,q,n)$ with $(s,r,q)\in\mathcal C_C(n)$ and the participation equation $n=NF(\pi(s,r,n))$; its value is $\Pi^*_C$.

The paper solves (13) in two steps. Step 1 fixes $n$, computes the maximal revenue $\Pi_C(n)$ over $\mathcal C_C(n)$, and among the revenue-maximizing allocations selects one that maximizes $\pi$, giving the per-driver profit $\pi_C(n)$. Step 2 solves $n_C=NF(\pi_C(n_C))$. The constants

$$n^C_1:=\bar S-(\Lambda_{21}-\Lambda_{12})t_{21},\qquad n^C_2:=\bar S+(\Lambda_{21}-\Lambda_{12})t_{12}$$

split capacities into scarce, moderate and ample zones.

Formalization targets

Goal: Corollary 1

$$\pi_C \text{ continuous and nonincreasing on }(0,\infty),\ \text{strictly decreasing on }(n^C_1,\infty);\qquad \exists!\,n_C>0:\ n_C=NF(\pi_C(n_C));$$ $$\pi_C(n)=\max_{(s,r,q)\in\mathcal C_C(n)}\pi(s,r,n)\quad(0<n\le N);\qquad \Pi^*_C=\Pi_C(n_C),\ \text{attained by the two-step tuple at } n_C .$$

Milestones

1. Proposition 1(1)–(3): the selected optimal allocation in each zone — $\bar s=n$, $r=q=0$ for $n\le n^C_1$; $\bar s+r_{12}=n$ with $r_{12}=\frac{t_{12}}{t_{12}+t_{21}}(n-n^C_1)$, $r_{21}=0$, $q=0$ for $n^C_1<n\le n^C_2$; $\bar s=\bar S$, $r_{12}=n^C_2-\bar S$, $r_{21}=0$, $\bar q=n-n^C_2$ for $n>n^C_2$.
2. Display (18): the closed form of $\pi_C(n)$ in the three zones.
3. Lemma 1 at $X=C$: if $\pi_C$ is driver-optimal (16) and nonincreasing, the two-step solution solves (13).

Significance

The corollary is what makes centralized control a usable benchmark. It says the platform's joint problem, which couples a linear program with a fixed-point equation in $n$, decomposes: the platform can optimize a fixed capacity and then let drivers settle, with no loss. Under the other regimes of the paper this alignment of platform and driver incentives can fail, and the paper's comparisons of regimes (value of admission control, value of repositioning) are measured against the regime-$C$ optimum characterized here. Proposition 1 also makes explicit that a centralized platform keeps drivers idle only when all demand is served.

The paper's proofs are in its e-companion, which is not part of the source this mission cites; no machine-checked version of these results exists. The mission produces a checked parametric solution of a small linear program, a checked closed form of the per-driver profit, and a checked equilibrium-and-optimality argument, with every standing assumption made explicit.

Difficulty

Proposition 1 is a parametric linear program in eight variables, but stating its solution is not enough: each zone needs an upper bound on $\bar s$ valid on all of $\mathcal C_C(n)$, and a second optimization (per-driver profit) over the set of revenue maximizers, which is not a singleton. Driver-optimality (16) is a separate claim about a different objective over the full set $\mathcal C_C(n)$; it holds only because Assumption 2 makes the capacity used for excess cross-location demand still profitable for drivers. Finally, Lemma 1 compares tuples of (13) at different capacities, which requires monotonicity of $\Pi_C$ and $\pi_C$ in $n$ and the participation equation at both capacities. The natural reading of "$n^*_C=n_C$" as uniqueness of the optimal capacity is false when $n_C>n^C_2$, since wasteful repositioning can keep revenue $\gamma p\bar S$ at a smaller capacity; the targets assert attainment instead.

Formalization scope

All objects live in the namespace RideHailing.Central, in one definition file. Routes are named fields (Λ11 … Λ22, t11 … t22), not indexed. $F$ is ProbabilityTheory.cdf μ of a measure μ on $\mathbb R$. The standing hypothesis Model.Standing bundles $0<\gamma<1$, $\bar\gamma p-c>0$, Assumptions 1–3 as printed, and the paper's implicit conventions, made explicit: $t_{lk}>0$, $\Lambda_{11},\Lambda_{22}\ge0$, $p>0$, $N>0$, $c>0$, $\mu$ a probability measure with continuous CDF.

$\Pi_C(n)$ and $\pi_C(n)$ are defined as suprema over the allocation problem; every target states attainment with IsGreatest or with membership in the selected set, so no conclusion rests on a supremum convention. Capacities in (13) and (16) are strictly positive, because (12d) divides by $n$.

Trivializing formalizations are ruled out: $\pi_C$ is defined through the two-step selection, never by its closed form (18); "max" is never an unattained supremum; feasibility for (13) includes the participation equation and $n>0$; and $n_C$ is characterized by $\exists!$ and then quantified over, never chosen.

Regimes $M$ and $A$ of the paper, which need objects defined only in the e-companion, are out of scope, as are the reformulation of §2.4 and Remark 2. Proofs of any milestone, and lemmas on monotonicity of $\Pi_C$ or attainment of linear programs over $\mathcal C_C(n)$, are welcome.

Selected references

• P. Afèche, Z. Liu, C. Maglaras, Ride-Hailing Networks with Strategic Drivers: The Impact of Platform Control Capabilities on Performance, Manufacturing & Service Operations Management, Articles in Advance, 2023. https://doi.org/10.1287/msom.2023.1221