1

1

Introduction

This document contains several supplemental results for our paper; it’s not self-contained and assumes a degree of familiarity with the main paper’s appendix. It’s organized as follows: • In Section 2, we give two additional examples of our game with nonmonotone earnings. • In Section 3, we show that the game from the main text is formally equivalent to a game where the firms choose offers, instead of values, and we illustrate this equivalence with two examples. • In Section 4, we show that the game from the main text is isomorphic to a game where the firms have payoff-relevant types that are unobserved by the consumer. • In Section 5, we examine the relationship between the more discerning order the classical information orders of Blackwell and Lehmann. We find that the more discerning order is generally independent of these orders, though there are important cases where they overlap. • In Section 6, we extend the strongly more discerning order to asymmetric signals and show that Propositions 5 to 7 continue to hold. • In Section 7, we examine a special case of our game and develop a joint condition on the signals and the firms’ earnings under which more discernment leads to a worse offer by one firm, and we illustrate this with two examples.

2

Additional Examples

In this section, we give two additional examples to help illustrate our game and the effects of more discernment. Both are of interest as the firms have nonmonotone earnings, and so illustrate, in a very concrete way, that our results do not depend on the monotonicity of the firms’ earnings. The first example is of additional interest because it shows that restriction of σ ∈ (0, 1/2) in Example 1 is completely without loss. Example OA1. A Variant of Example 1. The firms are symmetric. In particular, VA = VB = [0, 1] and wA (v) = wB (v) = v − v 2 . The consumer has the same signal as in Example 1, save that, σ ∈ R++ . For each σ > 0, standard arguments give that there is a unique MCU equilibrium ? (vA (σ), vB? (σ), φ? (σ)). Since the firms are symmetric, vA? (σ) = vB? (σ), so we write v ? (σ) for the value the firms offer. √ √ Claim OA1. We have v ? (σ) = 21 (1 − 2πσ 2 + 2πσ 2 + 1), which is decreasing in σ. Thus, more discernment is good for the consumer and bad for the firms.

2

S14

Firm A’s Value 0.512

Firm B’s Value 0.253

Pr(Consumer Chooses A’s Offer) 0.513

Consumer’s Payoff 0.386

Firm A’s Payoff 0.128

Firm B’s Payoff 0.061

S24

0.524

0.257

0.527

0.397

0.131

0.059

S34

0.634

0.385

0.841

0.594

0.195

0.014

S44

0.630

0.444

0.933

0.618

0.217

0.003

Signal

Table 1: Example OA2 – Description of Each Signal’s MCU Equilibrium Proof of Claim OA1. Focus on firm A in the reduced game. Since wA (v) = v − v 2 and B mA (vA , vB ) = 12 (Pr(S1 |vA , vB ) + Pr(S2 |vB , vA )) = Φ( vA −v ), A’s reduced game payoff is σ vA −vB R 2 πA (vA , vB ) = (vA − vA )Φ( σ ). Since it’s always optimal for A (and B by symmetry) ∂π R (v ,v ) to choose an interior best response,1 the first order condition binds. Thus, A∂vAA B = B B ) + (vA − vA2 )Φ0 ( vA −v ) = 0, where Φ0 (x) is the standard normal density. (1 − 2vA )Φ( vA −v σ σ Since vA? (σ) = vB? (σ) = v ? (σ) in the MCU equilibrium, we have (1 − 2v ? (σ))/2 + (v ? (σ) − √ √ √ v ? (σ)2 )/(σ 2π) = 0 by Lemma 3. A bit of algebra gives v ? (σ) = 21 (1 − 2πσ 2 + 2πσ 2 + 1), which is easily seen to be decreasing in σ. 4 Example OA2. A Variant of Example 4. Let VA = [0, 1], VB = [0, 1/2], wA (v) = v − v 2 , and wB (v) = v − 2v 2 . Consider four signals for the consumer S14 , S24 , S34 , and S44 , which are from the family considered in Example 1 and differ in discernment. Specifically, S14 has noise with standard deviation σS14 = 8 and is the least discerning, S24 has noise with standard deviation σS24 = 4 and is the second least discerning, S34 has noise with standard deviation σS34 = 1/4 and is the second most discerning, and S44 has noise with standard deviation σS44 = 1/8 and is the most discerning. Computation gives Table 1, which describes each firm’s (MCU) equilibrium value, the equilibrium probability that the consumer chooses firm A’s offer, and the players’ (ex-ante) equilibrium payoffs for each signal. The table shows that more discernment improves the consumers’ payoff, harms the firms, and that the offers are nonmonotone. 4

3

Offers Instead of Values

In the main text, we claimed that it’s without loss to require the firms to pick values instead of offers. Our goal, in this section, is to prove this claim. To these ends, we first introduce an “offer game,” make a few observations, and then give the proof. 1

Values of 0 and 1 are never best responses for firm A. If it set a value of 0 or 1, then it’s payoff would be zero, whereas it would have a strictly positive payoff if it set a value of 1/2 because wA (1/2) = 1/4 and because S has a strictly positive conditional mass function.

3

Offer Game In the offer game, the consumer and firms interact in the following three stages. Stage 1

Each firm i ∈ {A, B} chooses an offer oi ∈ Oi , where Oi is the firm’s set of possible offers. Each offer oi may specify a number of characteristics – e.g., premiums and reimbursement rates in the case of insurance plans, portfolio composition and fees in the case of retirement plans, or even price and quality in the case of bananas. These characteristics determine the value of oi to the consumer and the firm. We denote the consumer’s value of oi by vˆi (oi ) and the firm’s value by wˆi (oi ), where vˆi : Oi → R and wˆi : Oi → R.

Stage 2

Nature relabels the firms’ offers in the standard fashion – both have an equal chance of being in the first position. We label the firm whose offer is first 1 and the other firm 2.

Stage 3

The consumer examines the first and second offers (o1 , o2 ). As usual, her examination only gives a (noisy) signal Sˆ about the offers’ values (ˆ v1 (o1 ), vˆ2 (o2 )). (Notice (ˆ v1 (o1 ), vˆ2 (o2 )) = (ˆ vA (oA ), vˆB (oB )) if A is first and (ˆ v1 (o1 ), vˆ2 (o2 )) = (ˆ vB (oB ), vˆA (oA )) if B is first.) The signal has countable support S and a strictly positive conditional mass P 0 function fSˆ (s|v, v 0 ), where fSˆ : S × Vˆ 2 → R++ so that s∈Sˆ fSˆ (s|v, v ) = 1 for each (v, v 0 ) and that fSˆ (s|v, v 0 ) is continuous in (v, v 0 ) for each s and where Vˆ = ∪i∈{A,B} vˆi (Oi ) is the set of possible values the consumer could receive. After she sees her signal, the consumer chooses either the first or the second offer. If she chooses the first offer (second offer), then she earns vˆ1 (o1 ) (earns vˆ2 (o2 )), the first firm earns wˆ1 (o1 ) (second firms earns wˆ2 (o2 )), and the other firm earns zero.

A (pure) strategy for each firm is simply an offer, and a strategy for the consumer is a map φ : S → {1, 2} that, for each signal s ∈ S, tells her whether to take the first or second offer. We write (oA , oB , φ) for the vector of players’ strategies. Since the consumer chooses firm i’s offer oi if (i) it’s first and she sees a signal s such that φ(s) = 1 or (ii) it’s second and she sees a signal s such that φ(s) = 2, firm i’s payoff is π ˆi (oA , oB , φ) =

wˆi (oi ) X ( 2

fS (s|ˆ vi (oi ), vˆ−i (o−i )) +

{s|φ(s)=1}

X

fS (s|ˆ v−i (o−i ), vˆi (oi ))).

{s|φ(s)=2}

As to the consumer, before she receives her signal, there’s a half chance that firm A’s offer is first and firm B’s offer is second and a half chance of the reverse. Thus, 4

the first offer’s value is a random variable Vˆ1 and the second offer’s value is a random variable Vˆ2 with joint mass function fVˆ1 Vˆ2 (v, v 0 ), where fVˆ1 Vˆ2 (v, v 0 ) = 1/2 if (v, v 0 ) ∈ Q = {(ˆ vA (oA ), vˆB (oB )), (ˆ vB (oB ), vˆA (oA ))} and fVˆ1 Vˆ2 (v, v 0 ) = 0 otherwise. Once she observes her signal s ∈ S, she computes the posterior distribution of (Vˆ1 , Vˆ2 ) using Bayes’ rule and obtains 0

0

fSˆ (s|v,v0 )

f ˆ (s|v, v )fV1 V2 (v, v ) 2fSˆ (s) = fVˆ1 Vˆ2 |S (v, v |s) = S 0 fSˆ (s) 0

if (v, v 0 ) ∈ Q else,

P where fS (s) = (v,v0 )∈V 2 fS (s|v, v 0 )fV1 V2 (v, v 0 ). Thus, the expected values of the first and second offers are fSˆ (s|ˆ vA (oA ), vˆB (oB )) f ˆ (s|ˆ vB (oB ), vˆA (oA )) + vˆB (oB ) S 2fSˆ (s) 2fSˆ (s) f ˆ (s|ˆ vA (oA ), vˆB (oB )) fS (s|ˆ vB (oB ), vˆA (oA )) E(Vˆ2 |oA , oB , s) = vˆB (oB ) S + vˆA (oA ) . 2fSˆ (s) 2fSˆ (s) E(Vˆ1 |oA , oB , s) = vˆA (oA )

(1) (2)

Hence, after observing her signal, her interim payoff to choosing offer j ∈ {1, 2} is E(Vˆj |oA , oB , s). P It follows that her payoff to (oA , oB , φ) is π ˆC (oA , oB , φ) = s∈S E(Vˆφ(s) |vA , vB , s)fS (s). An offer equilibrium is a strategy vector (o?A , o?B , φ? ) such that all three players maximize their payoffs taking the strategy of the others as given, i.e., such that (i) π ˆA (o?A , o?B , φ? ) ≥ π ˆA (oA , o?B , φ? ) for all oA ∈ OA , (ii) π ˆB (o?A , o?B , φ? ) ≥ π ˆB (o?A , oB , φ? ) for all oB ∈ OB , and ˆC (o?A , oB , φ) for all φ ∈ Φ. A minimal consumer undominated (iii) π ˆC (o?A , o?B , φ? ) ≥ π (MCU) offer equilibrium is an offer equilibrium (o?A , o?B , φ? ) such that (i) the consumer’s strategy φ? is undominated and (ii) the values of the firms’ offers are lower than in any other offer equilibrium where the consumer plays an undominated strategy, i.e., (ˆ vA (o?A ), vˆB (o?B )) ≤ (ˆ vA (o0A ), vˆB (o0B )) for any offer equilibrium (o0A , o0B , φ0 ) where φ0 is undominated. Equivalency With the preliminaries out of the way, we can give our equivalence result. We’ll start with the offer game, “induce” an associated main game via the appropriate choice of its primitives – which are VA , VB , wA (v), wB (v), and the consumer’s signal S – and then show that the equilibria of both games are (essentially) the same.2 We need the following regularity condition. Regularity Condition I. The following all hold: 1. The consumer’s signal Sˆ strictly points and is log-supermodular. 2. The set of values each firm can offer the consumer is a compact interval, i.e., for 2

An alternative, equally valid approach, would be to start with the main game, find primitives on the offer game that “rationalize” the main game, and then show that both equilibria are equivalent.

5

each firm i, vˆi (Oi ) = [ˆv i , vˆi ] for some 0 ≤ vˆi < vˆi < ∞. Let Vˆi = [ˆv i , vˆi ] and note Vˆ = VˆA ∪ VˆB . 3. If firm i wishes to offer a value v ∈ Vˆi , then it’s maximum earnings are achieved at some feasible offer, i.e., the solution to max{wˆi (o)|o ∈ Oi and vˆi (o) = v} exists for each v ∈ Vˆi . Let w˜i (v) = max{wˆi (o)|o ∈ Oi and vˆi (o) = v} and let Hi (v) = arg max{wˆi (o)|o ∈ Oi and vˆi (o) = v}. The first regularity condition is standard. The second regularity condition ensures the offer game can be mapped to a main game (as the main game assumes each firm can offer values from an interval). The last regularity condition is a technical one that concerns the firms’ problems. In general, if firm i wants to offer the consumer a value of v, then there are many offers it can choose, each of which provides it with different earnings. This condition says that the supremum amount i can earn for any such offer is obtained by some offer. With the regularity condition in place, we induce a main game by setting VA = VˆA , wA (v) = w˜A (v), VB = VˆB , wB (v) = w˜B (v), and S = Sˆ (i.e., setting fS (s|v, v 0 ) = fSˆ (s|v, v 0 ) for all (v, v 0 ) ∈ V 2 = Vˆ 2 and all s ∈ S).3 We call this the associated main game. Proposition OA1. Equivalency of Offers and Values. If Regularity Conditions I hold, then an offer game strategy profile (o?A , o?B , φ? ) is a MCU offer equilibrium if and only if (o?A , o?B , φ? ) ∈ HA (vA? ) × HB (vB? ) × {φU }, where (vA? , vB? , φ? ) is the MCU equilibrium of the associated main game. It follows that the players “behave” in the same manner in both the MCU equilibrium and the MCU offer equilibrium, i.e., that (i) that each firm i offers the consumer the same value in both equilibria, (ii) that the consumer chooses i with the same probability in both equilibria, and (iii) that the consumer and the firms earn the same in both equilibria (since wA (vA? ) = wˆA (o?A ) and wB (vB? ) = wˆB (o?B )). Since our results concern only the values of the firms’ offers, the probability the consumer chooses a firm, and payoffs and welfare, Proposition OA1 gives that each of our results holds for all MCU offer equilibria. For example, if we increase the consumer’s signal in the more discerning order, then Proposition 4 gives that, in every MCU offer equilibrium, the offers become more competitive when the firms are symmetric, i.e., VˆA = VˆB and w˜A (v) = w˜B (v). Thus, it’s without loss to focus on the simpler main game and abstract away from the offers’ particularities. The intuition is that the consumer has the same signal Sˆ in both games. Thus, in both, she estimates that A’s offer (B’s offer) is better after a signal in Sˆ1 (in Sˆ2 ) – see Lemma OA1, below, and Proposition 1. It follows that, in both games, (i) she plays φU and (ii) Pr(Sˆ1 |v, v 0 ) (Pr(Sˆ2 |v, v 0 )) is the probability the signal indicates the first offer (second offer) is better. In Since S and Sˆ share a common support S, they’re the same when they have the same conditional ˆ distribution function. Also, V = VA ∪ VB = VˆA ∪ VˆB = V. 3

6

the offer game, result (i) implies that the firms only care about (a) their offers’ values (since these determine the probabilities they’re chosen) and (b) their earnings from their offers. Since an optimizing firm i always chooses an offer that maximizes its earnings subject to the value it provides, it chooses an offer o such that (vi (o), wi (o)) solves w˜i (vi (o)) = wi (o). Thus, it’s tradeoff between its earnings and the value it provides to the consumer is completely described by w˜i (v). Since it faces exactly the same tradeoff between value and earnings in the associated main game, it behaves in the same way in both games with respect to the value of its offer. Hence, the equilibria of both games are equivalent. Result (ii) ensures that, in the offer game, the more discerning order continues to model increases in discernment. The naive concern is that, in changing the economic environment, we may have changed the consumer’s problem to such a degree that a (joint) increase in Pr(Sˆ1 |v, v 0 ) and Pr(Sˆ2 |v 0 , v) for v > v 0 no longer increases discernment, i.e., the probability her signal correctly indicates the better offer. Fortunately, this is not the case as (ii) ensures that an increase in Pr(Sˆ1 |v, v 0 ) and Pr(Sˆ2 |v 0 , v) for v > v 0 actually increases the probability the signal indicates the first (second) offer is better, when it’s actually better, and so increases the probability the signal correctly indicates the better offer.4 We illustrate the proposition with two examples. Example OA3. An Equivalence of Offer Equilibria and the MCU Equilibria. Recall, from footnote 20 of the main text, that we could rationalize the primitives of the Example 1 with a homogenous product duopoly, a la Nermuth [4] and Perloff and Salop [5]. Let’s take this rationalization seriously and show that the MCU offer equilibrium of the duopoly game is equivalent the MCU equilibrium we previously found. Suppose the firms A and B make a homogenous product at zero marginal cost and offer prices pA ∈ [0, 1] and pB ∈ [0, 1], respectively, to the consumer. The consumer has unit demand, values the good at one, and has utility that is quasi-linear in money. Once the consumer chooses an offer, she purchases a unit from the firm that made the offer at the offered price. Thus, if she selects, say, firm A’s offer, then she gets a value of vˆA (pA ) = 1 − pA and A earns wˆA (pA ) = pA . Likewise, if she chooses B’s offer, she gets a value of vˆB (pB ) = 1 − pB and B earns wˆB (pB ) = pB . For simplicity, the consumer has the same signal as in Example 1 with σ = 1/4. If the firms set prices (pA , pB ), then they provide values (1 − pA , 1 − pB ) to the consumer. 4

If we think of the “better offer” as the better of the relabeled offers (o1 , o2 ), then the probability S correctly indicates it is Pr(Sˆ1 |v1 (o1 ), v2 (o2 )) when v1 (o1 ) > v2 (o2 ), is Pr(Sˆ2 |v1 (o1 ), v2 (o2 )) when v1 (o1 ) < v2 (o2 ), and is 1 when v1 (o1 ) = v2 (o2 ). If we think of the “better offer” as the better of the firms’ offers, then the probability S correctly indicates it is 21 (Pr(Sˆ1 |vA (oA ), vB (oB )) + Pr(S2 |vB (oB ), vA (oA ))) when vA (oA ) > vB (oB ), is 1 2 (Pr(S1 |vB (oB ), vA (oA )) + Pr(S2 |vA (oA ), vB (oB ))) when vA (oA ) < vB (oB ), and is 1 when vA (oA ) = vB (oB ). In both interpretations, S S 0 implies S 0 correctly indicates the better offer more often than S.

7

So the probabilities their offers are chosen when the consumer follows φU are m ˆ A (pA , pB ) = mA (ˆ vA (pA ), vˆB (pB )) = Φ(4(pB − pA )) and m ˆ B (pA , pB ) = Φ(4(pA − pB )), respectively. Thus, their respective earnings in “the reduced game” are π ˆAR (pA , pB ) = pA Φ(4(pB − pA )) and π ˆBR (pA , pB ) = pB Φ(4(pA − pB )). Hence, an argument analogous to the Proof of Claim 1 pπ , so a MCU offer equilibrium is (p?A , p?B , φ? ) = gives that the firms set prices p?A = p?B = 32 pπ pπ , 32 , φU ) as the consumer always follows φU by Proposition OA1. ( 32 Claim 1 gives that the unique MCU equilibrium of Example 1 is (vA0 , vB0 , φ0 ) = (1 − pπ pπ , 1 − 32 , φU ). Since vˆA (pA ) = 1 − pA and vˆB (pB ) = 1 − pB , we have (ˆ vA (p?A ), vˆB (p?B ), 32 φU ) = (vA0 , vB0 , φU ), i.e., the equilibria are equivalent. The reason for this equivalency is that Proposition OA1 holds. Specifically, the firms can offer the same values, the consumer’s signal is the same, and if a firm, say A, wants to offer the consumer a value of vA , then it must do so by charging a price of pA = 1 − vA , so w˜A (vA ) = 1 − vA , which is exactly its earnings Example 1. (Likewise, w˜B (vB ) = 1 − vB .) 4 Before proceeding, we wish to point out that, in the previous example, w˜A (v) and w˜B (v) trace out the Pareto frontier between the consumer and firms A and B respectively. Indeed, for each (v, w˜i (v)), it’s impossible to increase the consumer’s payoff v without decreasing firm i’s payoff w˜i (v) and vice-versa when w˜i (v) is strictly decreasing. Thus, in the example’s MCU offer equilibrium, the trade that occurs between the consumer and the firm she chooses is Pareto efficient. More generally, the trade that occurs in any MCU offer equilibrium between the consumer and the firm she chooses is also Pareto efficient when w˜A (v) and w˜B (v) are strictly decreasing. Hence, so to is the MCU equilibrium trade by Proposition OA1. The next example illustrates how one can use Proposition OA1 to indirectly solve for an offer equilibrium. Example OA4. Solving for a MCU Offer Equilibrium. Suppose that firms A and B compete on the basis of price and quality. Specifically, they offer prices pA ∈ [0, 1] and pB ∈ [0, 1], respectively, and offer qualities qA ∈ [0, 1/2] and qB ∈ [0, 1/2], respectively. For simplicity, the firms have a constant marginal cost of quality of 1/2 and receive a subsidy of 1/4.5 The consumer has a value of 1 + 2q for a product of quality q and has utility that is quasi-linear in money. Thus, if she selects, say, A’s offer, she gets a value of vˆA (pA , qA ) = 1 + 2qA − pA and A earns wˆA (pA , qA ) = 41 + pA − 12 qA . Likewise, if she chooses B’s offer, then she gets a value of vˆB (pB , qB ) = 1 + 2qB − pB and B earns wˆB (pB , qB ) = 41 + pB − 12 qB . For simplicity, the consumer has the same signal as in Example 1 with σ = 1/4. Let’s construct the corresponding main game. First, notice that the firms can offer the 5

The subsidy is a simplifying device only; it saves on notation by ensuring that the firms’ earnings are always weakly positive.

8

consumer a value in [0, 2], so set VA = VB = [0, 2]. Second, let the consumer have the same signal. Third, note that if A wants to offer the consumer a value of vA , then it can choose any (pA , qA ) ∈ {(p, q)|1 + 2q − p = vA }. Of these pairs, it picks the one that maximizes its payoff. Specifically, it chooses (p?A , qA? )(vA ) =

(1, vA ) 2

if vA ≤ 1

(2 − v , 1 ) if v > 1. A 2 A

In other words, HA (v) = (p?A , qA? )(vA ). Thus, A’s earnings as a function of the consumer’s value vA is 54 − 14 vA if vA ≤ 1 and is 2 − v if v > 1, so set 5 − 1v A wA (vA ) = 4 4 2 − v

if vA ≤ 1 if v > 1.

Analogously, set wB (vB ) = 1/2 − vB , so HB (v) = HA (v). The main game, so defined, meets our regularity conditions, implying that the MCU equilibrium we’ll find also defines a MCU offer equilibrium per Proposition OA1. Numerical methods give that the MCU equilibrium is (vA? , vB? , φ? ) = (1.687, 1.687, φU ). Hence, Proposition OA1 gives that a vector (p?A , qA? , p?B , qB? , φ? ) is an offer equilibrium only if (p?A , qA? ) ∈ HA (1.687), (p?B , qB? ) ∈ HB (1.687), and φ? = φU . We thus have that p?A = p?B = 2 − 1.687 = 0.313 and qA? = qB? = 1/2. 4 We need two lemmas to prove the proposition. Lemma OA1. The Offers’ Expected Values. If the firms play (oA , oB ) and Regularity Condition I holds, then a signal s1 ∈ S1 implies that E(Vˆ1 |oA , oB , s1 ) ≥ E(Vˆ2 |σA , σB , s1 ) and a signal s2 ∈ S2 implies that E(Vˆ1 |oA , oB , s2 ) ≤ E(Vˆ2 |oA , oB , s2 ). Proof. Analogous to the proof of Proposition 1. Let’s focus on the case of s1 ∈ S1 . By equations (1) and (2), E(Vˆ1 |oA , oB , s1 ) ≥ E(Vˆ2 |oA , oB , s1 ) if and only if (vA (oA ) − vB (oB ))(fS (s1 |vA (oA ), vB (oB )) − fS (s1 |vB (oB ), vA (oA ))) ≥ 0. If vA (oA ) = vB (oB ), then the inequality clearly obtains. If vA (oA ) > vB (oB ), strict pointing gives fS (s1 |vA (oA ), vB (oB )) − fS (s1 |vB (oB ), vA (oA )) > 0, so the inequality obtains. Likewise, if vA (oA ) < vB (oB ), strict pointing gives fS (s1 |vA (oA ), vB (oB )) − fS (s1 |vB (oB ), vA (oA )) < 0, so the inequality holds. Since the argument is analogous for s2 ∈ S2 , the lemma obtains. Lemma OA2. The Consumer’s Undominated Strategy. 9

If Regularity Condition I holds, then the consumer’s unique undominated strategy in the offer game is φU . Proof. By Lemma OA1, φU is undominated as it implements the consumer’s best response. The argument for uniqueness parallels the Proof of Proposition 2. To sketch it, suppose there’s another undominated strategy φ0 . Since both generate the same payoff to the consumer for all (oA , oB ) and yet differ, it must be that there’s a signal s after which E(Vˆ1 |oA , oB , s) = E(Vˆ2 |oA , oB , s) for all (oA , oB ). This happens if and only if (vA (oA ) − vB (oV ))(fSˆ (s|vA (oA ), vB (oB )) − fSˆ (s|vB (oB ), vA (oA ))) = 0. Since we can always choose (oA , oB ) such that vA (oA ) − vB (oV ) 6= 0, this inequality only obtains if fSˆ (s|v, v 0 ) = fSˆ (s|v 0 , v) for all (v, v 0 ) ∈ Vˆ 2 , which is exactly what strict pointing rules out. Proof of Proposition OA1. Straightforward, but tedious. Before we begin, we need six preliminary facts. Fact 0 The consumer’s unique undominated strategy in the offer game and the associated main game is φU . Proof. Immediate from Lemma OA2 and Proposition 2. Fact 1 If (vA , vB , φU ) a joint strategy in the associated main game, then πA (vA , vB , φU ) = π ˆA (oA , oB , φU ) πB (vA , vB , φU ) = π ˆB (oA , oB , φU ) for any oA ∈ HA (vA ) and oB ∈ HB (vB ). Proof. Without loss, we focus on firm A. Since vˆA (oA ) = vA and vˆB (oB ) = vB , we have πA (vA , vB , φU ) = wA (vA )mA (ˆ vA (oA ), vˆB (oB )). Since wA (vA ) = wˆA (oA ), we have πA (vA , vB , φU ) = wˆA (oA )mA (ˆ vA (oA ), vˆB (oB )). Since the consumer’s follows φU by Fact 0, the probability she chooses A’s offer is mA (ˆ vA (oA ), vˆB (oB )) since the signal is the same in both games. Thus, we have πA (vA , vB , φU ) = π ˆA (oA , oB , φU ). Fact 2 If (oA , oB , φU ) is a strategy vector in the offer game such that oA ∈ HA (ˆ vA (oA )) and oB ∈ HB (ˆ vB (oB )), then π ˆA (oA , oB , φU ) = πA (ˆ vA (oA ), vˆB (oB ), φU ) π ˆB (oA , oB , φU ) = πB (ˆ vA (oA ), vˆB (oB ), φU ). Proof. Without loss, we focus on firm A. Let vA = vˆA (oA ) and vB = vˆB (oB ). 10

Since oA ∈ HA (vA ), we have wˆA (oA ) = wA (vA ) by construction. Thus, as the consumer follows φU and has the same signal in both games, we have π ˆA (oA , oB , φU ) = wˆA (oA )mA (ˆ vA (oA ), vˆB (oB )) = wA (vA )mA (vA , vB ) = πA (vA , vB , φU ). Fact 3 If (o?A , o?B , φU ) is an offer equilibrium, then o?A ∈ HA (ˆ vA (o?A )) and o?B ∈ HB (ˆ vB (o?B )). Proof. Without loss, we focus on firm A. Let vA? = vˆA (o?A ) and vB? = vˆB (o?B ). For o?A and o0A ∈ HA (vA? ),6 we have wˆA (oA ) ≤ wˆA (o0A ) = wA (vA? ) by optimality and construction. If wˆA (o?A ) < wˆA (o0A ), then A does strictly better by offering o0A instead of o?A , i.e., π ˆA (o?A , o?B , φU ) = wˆA (o?A )mA (vA? , vB? ) < wˆA (o0A )mA (vA? , vB? ) = π ˆA (o0A , o?B , φU ). This contradicts the fact that (o?A , o?B , φU ) is an offer equilibrium. Hence, we must have wˆA (o?A ) = wA (vA? ), equivalently, o?A ∈ HA (vA? ). Fact 4 If (o?A , o?B , φU ) is an offer equilibrium, then (ˆ vA (o?A ), vˆB (o?B ), φU ) is a MCU equilibrium (in the associated main game). Proof. We argue by contradiction. If (ˆ vA (o?A ), vˆB (o?B ), φU ) is not a equilibrium, then one firm, say A, must do strictly better in the associated main game by playing v 0 6= vˆA (o?A ), i.e., πA (v 0 , vˆB (o?B ), φU ) > πA (ˆ vA (o?A ), vˆB (o?B ), φU ). Let o0A ∈ HA (v 0 ), then Fact vB (o?B )) by Fact 3) and ˆA (o0A , o?B , φU ) (since o?B ∈ HB (ˆ 1 gives πA (v 0 , vˆB (o?B ), φU ) = π ˆA (o0A , o?B , φU ) > ˆA (o?A , o?B , φU ). Thus, π Facts 2 and 3 give πA (ˆ vA (o?A ), vˆB (o?B ), φU ) = π π ˆA (o?A , o?B , φU ), i.e., A has incentive to defect from (o?A , o?B , φU ) in the offer game, a contradiction. Fact 5 If (vA? , vB? , φU ) is a MCU equilibrium, then any (o?A , o?B , φU ) such that o?A ∈ HA (vA? ) and o?B ∈ HB (vB? ) is an offer equilibrium. Proof. We argue by contradiction. If (o?A , o?B , φU ) is not an offer equilibrium, then one firm, say A, does strictly better in the offer game by playing o0 6= o?A , i.e., vA (o0 )), then wˆA (o0 ) ≤ wˆA (o00 ) = ˆA (o?A , o?B , φU ). Let o00 ∈ HA (ˆ π ˆA (o0 , o?B , φU ) > π ˆA (o0 )mA (ˆ vA (o0 ), vB? ) ≤ wA (ˆ vA (o0 )) by optimality, so we have that π ˆA (o0 , o?B , φU ) = w wˆA (o00 )mA (ˆ vA (o00 ), vB? ) = π ˆA (o00 , o?B , φU ), implying π ˆA (o00 , o?B , φU ) > π ˆA (o?A , o?B , φU ). Since (i) Fact 2 gives π ˆA (o00 , o?B , φU ) = πA (ˆ vA (o0 ), vB? , φU ) (because (a) o00 ∈ HA (ˆ vA (o0 )) implies o00 ∈ HA (ˆ vA (o00 )) as vˆA (o00 ) = vˆA (o0 ) and (b) because o?B ∈ HB (vB? ) implies gives o?B ∈ HB (ˆ vB (o?B ))) and (ii) Fact 1 gives π ˆA (o?A , o?B , φU ) = πA (vA? , vB? , φU ), we have πA (vA (o00 ), vB? , φU ) > πA (vA? , vB? , φU ). Thus, firm A has incentive to defect from (vA? , vB? , φU ) in the associated main game, a contradiction. This concludes the list of preliminary facts. With these facts in hand, it’s easy to prove the proposition. Recall that (vA? , vB? , φ? ) is the unique MCU equilibrium (of the associated main game) per Proposition 3. We first show that any (o?A , o?B , φ? ) ∈ HA (vA? ) × HB (vB? ) × {φU } is a MCU offer equilibrium. Since 6

The existence of o0A follows from part three of Regularity Condition I.

11

φ? = φU by Fact 0, we have (o?A , o?B , φU ) is an offer equilibrium by Fact 5. All that remains to do is show that no offer equilibrium where the consumer plays her undominated strategy has strictly smaller values than (o?A , o?B , φ? ). We argue by contradiction. If not, then there’s another offer equilibrium (o0A , o0B , φ0 ), where φ0 is undominated and either vˆA (o0A ) < vˆA (o?A ) or vˆB (o0B ) < vˆB (o?B ). Since Fact 0 gives φ0 = φU , Fact 4 implies that (ˆ vA (o0A ), vˆB (o0B ), φU ) is an equilibrium with vA (o0A ) < vA? or vB (o0B ) < vB? . Hence, (vA? , vB? , φ? ) isn’t the MCU equilibrium, a contradiction. As to the converse, we’ll next show that: if (o?A , o?B , φ? ) is a MCU offer equilibrium, then (ˆ vA (o?A ), vˆB (o?B ), φ? ) is a MCU equilibrium. Since the MCU equilibrium is unique, we must have that (ˆ vA (o?A ), vˆB (o?B ), φ? ) = (vA? , vB? , φ? ). Since Facts 0 and 3 give that (o?A , o?B , φ? ) ∈ HA (ˆ vA (o?A )) × HB (ˆ vB (o?B )) × {φU } = HA (vA? ) × HB (vB? ) × {φU }, we obtain the proposition. It remains to show that if (o?A , o?B , φ? ) is a MCU offer equilibrium, then (ˆ vA (o?A ), vˆB (o?B ), φ? ) is a MCU equilibrium of the associated main game. Let vA? = vA (o?A ) and vB? = vB (o?B ). Fact 0 gives that (o?A , o?B , φ? ) = (o?A , o?B , φU ). Thus, Fact 4 gives that (ˆ vA (o?A ), vˆB (o?B ), φ? ) is an equilibrium. All that remains to do is show that there’s no equilibrium where the consumer plays her undominated strategy has smaller values than (ˆ vA (o?A ), vˆB (o?B ), φ? ). We argue by contradiction. If not, then there’s another equilibrium (vA0 , vB0 , φ0 ) where φ0 is undominated and either vA0 < vˆA (o?A ) or vB0 < vˆB (o?B ). Since Fact 0, we have (vA0 , vB0 , φ0 ) = (vA0 , vB0 , φU ). Let (o0A , o0B ) ∈ HA (vA0 ) × HB (vB0 ), then Fact 5 gives that (o0A , o0B , φU ) is an offer equilibrium with either vˆA (o0A ) < vˆA (o?A ) or vˆB (o0B ) < vˆB (o?B ). Hence, (o?A , o?B , φ? ) is not a MCU offer equilibrium, a contradiction.

4

Type-Uncertainty Instead of Nature’s Relabeling

In the main text, we claimed that the second stage makes our game isomorphic to one where consumer is uncertain of the firms’ types. Our goal, in this section, is to prove this. To these ends, we first introduce a “type-uncertainty game,” make a few observations, and then give the proof. Type-Uncertainty Game In the type-uncertainty game, the consumer and firms interact in the following three stages. Stage 1

Nature randomly assigns each firm i ∈ {A, B} a type θi ∈ {L, H}. Both types are equally likely and the assignment is independent of all other randomness.7

7

We use equally likely types for simplicity. It’s readily verified that this section’s results obtain so long as the types are independently and identically distributed.

12

A firm’s type θ determines its earnings function wˆθ (v), where wˆθ : Vˆθ → R and Vˆ = [ˆv , vˆ], with 0 ≤ vˆ < vˆ < ∞, is the common set of possible values.8 The firms observe each other’s types, while the consumer observes nothing about either firm’s type. Stage 2

Each firm i ∈ {A, B} simultaneously chooses a value vˆi ∈ Vˆ that its offer provides to the consumer. If i’s offer is selected, it earns wˆθi (ˆ vi ), else it gets zero.

Stage 3

The consumer receives firms’ offers and examines them. As usual, her examination only gives a (noisy) signal Sˆ about the offers’ values (ˆ vA , vˆB ). The signal has countable support S and a strictly positive conditional mass function P fSˆ (s|ˆ vA , vˆB ), where fSˆ : S × Vˆ 2 → R++ so that s∈Sˆ fSˆ (s|v, v 0 ) = 1 for each (v, v 0 ) and that fS (s|v, v 0 ) is continuous in (v, v 0 ) for each s. After she sees her signal, she chooses an offer. If she chooses A’s offer (B’s offer), then she earns vˆA (earns vˆB ), A earns wˆθA (ˆ vA ) (B earns wˆθB (ˆ vB )), and B earns zero (A earns zero).

A (pure) strategy for firm A is a map σA : Θ2 → Vˆ such that σA (θA , θB ) ∈ VˆθA for each (θA , θB ) ∈ Θ2 , i.e., is a map that specifies a feasible value for A for each pair of firm types. Likewise, a pure strategy for firm B is a map σB : Θ2 → Vˆ such that σB (θA , θB ) ∈ VˆθB for each (θA , θB ) ∈ Θ2 . And, for the consumer, a strategy is a map σC : S → {A, B} that, for each signal s ∈ S, tells her whether to take A’s or B’s offer. We write (σA , σB , σC ) for the joint strategy vector. We denote the players’ payoffs to a strategy vector (σA , σB , σC ) as π ˆA (σA , σB , σC ), π ˆB (σA , σB , σC ), and π ˆC (σA , σB , σC ). When firm A moves, it knows its own type θA and B’s type θB , and its earns wθA (σA (θA , θB )) if the consumer chooses it’s offer. Hence, its interim payoff is π ˆA (σA , σB , σC , θA , θB ) = wθA (σA (θA , θB ))

X

fSˆ (s|σA (θA , θB ), σB (θA , θB )).

{s|σC (s)=A}

It follows that π ˆA (σA , σB , σC ) = payoff is

1 ˆA (σA , σB , σC , θA , θB ). (θA ,θB )∈Θ2 4 π

P

π ˆB (σA , σB , σC , θA , θB ) = wθB (σB (θA , θB ))

X

Likewise, B’s interim

fSˆ (s|σA (θA , θB ), σB (θA , θB )),

{s|σC (s)=B}

so π ˆB (σA , σB , σC ) =

P

1 ˆB (σA , σB , σC , θA , θB ). (θA ,θB )∈Θ2 4 π

8

We use a common set of possible values for simplicity. It’s easily verified that this section’s results obtain when a firm’s type also determines its set of possible values, so long as this set is a compact interval.

13

As to the consumer, before she receives her signal, there’s a quarter chance that the offers are (σA (H, H), σB (H, H)), (σA (H, L), σB (H, L)), (σA (L, H), σB (L, H)), or (σA (L, L), σB (L, L)). Thus, A’s value is a random variable VˆA and B’s value is a random variable VˆB with joint mass function fVˆA VˆB (v, v 0 ), where 1/4 if (v, v 0 ) ∈ Q fVˆA VˆB (v, v 0 ) = 0 else, where Q = ∪(θA ,θB )∈Θ2 (σA (θA , θB ), σB (θA , θB )) is the set of possible values. Once the consumer observes her signal s ∈ S, she computes the posterior distribution of (VA , VB ) using Bayes’ rule and obtains 0 0 fS (s|v,v0 ) (v, v ) f (s|v, v )f ˆ ˆ ˆ S V V 4fSˆ (s) A B = fVˆA VˆB |Sˆ (v, v 0 |s) = 0 fSˆ (s)

if (v, v 0 ) ∈ Q else,

P 0 0 where fSˆ (s) = (v,v 0 )∈V 2 fSˆ (s|v, v )fVˆA VˆB (v, v ) is the probability of signal s. Thus, the expected values of VˆA and VˆB are E(VˆA |σA , σB , s) =

X (v,v 0 )∈Q

v

X fSˆ (s|v, v 0 ) f ˆ (s|v, v 0 ) and E(VˆB |σA , σB , s) = v0 S . 4fSˆ (s) 4fSˆ (s) 0 (v,v )∈Q

Hence, after observing her signal, the consumer’s interim payoff to choosing offer j ∈ {A, B} is E(Vˆj |σA , σB , s). It follows that her ex-ante payoff is to (σA , σB , σC ) is π ˆC (σA , σB , σC ) = P ˆ s∈S E(VσC (s) |σA , σB , s)fSˆ (s). We say the firms strategies are symmetric if σA (θA , θB ) = σB (θB , θA ) for each (θA , θB ) ∈ 2 Θ . We say that the consumer’s strategy σC is s-undominated if it’s undominated when the firms are required to play symmetric strategies. (Note that if σC is s-undominated, then it may still be dominated in general; yet, so long as the firms play symmetric strategies, the consumer can do no better that σC .) An symmetric type-uncertain equilibrium is a strategy vector (σA? , σB? , σC? ) such that all three players maximize their payoffs taking the strategy of the others as given, i.e., π ˆA (σA? , σB? , σC? ) ≥ π ˆA (σA , σB? , σC? ) for all other possible strategies σA , π ˆB (σA? , σB? , σC? ) ≥ π ˆB (σA? , σB , σC? ) for all possible strategies σB , and π ˆC (σA? , σB? , σC? ) ≥ π ˆC (σA? , σB? , σC ) for all possible strategies σC , and the firms strategies are symmetric. A minimal consumer s-undominated symmetric type-uncertain (MCUSTU) equilibrium is a symmetric equilibrium (σA? , σB? , σC? ) such that (i) the consumer’s strategy σC? is s-undominated and (ii) the values of the firms’ offers are lower than in any other symmetric equilib14

rium where the consumer plays an s-undominated strategy, i.e., (σA? (θA , θB ), σB? (θA , θB )) ≤ (σA0 (θA , θB ), σB0 (θA , θB )) for all (θA , θB ) ∈ Θ2 and any other symmetric equilibrium (o0A , o0B , σC0 ) where σC0 is s-undominated. Equivalency With the preliminaries out of the way, we can give our equivalency result. We’ll start with the type-uncertainty game, “induce” a family of main games that depend on the firms’ types, and then show that the each MCUSTU equilibrium corresponds to the MCU equilibria of this family. We need the following regularity condition. Regularity Condition II. Let the consumer’s signal Sˆ strictly point, be symmetric, and be log-supermodular.9 With the regularity condition in place, for each pair of firm types (θA , θB ) ∈ Θ2 , we induce ˆ wA (v) = wθ (v), wB (v) = wθ (v), and S = Sˆ (i.e., we a main game by setting VA = VB = V, A B ˆ set the consumer’s signal to be S). We refer to this main game the associated main game for types (θA , θB ). This game has a unique MCU equilibrium under Regularity Condition II (per Proposition 3). Proposition OA2. Equivalency of Stage 2 and Type-Uncertainty. If Regularity Condition II holds, then a type-uncertainty game strategy profile (σA? , σB? , σC? ) is a MCUSTU equilibrium if and only if (i) σC? (s) = A ⇐⇒ φU (s) = 1 and σC? (s) = B ⇐⇒ φU (s) = 2 for each s ∈ S and (ii) (σA? (θA , θB ), σB? (θA , θB )) = (vA? (θA , θB ), vB? (θA , θB )) for each (θA , θB ) ∈ Θ2 , where (vA? (θA , θB ), vB? (θA , θB )) denotes the firms’ values in the unique MCU equilibrium of the associated main game for types (θA , θB ). That is, in each MCUSTU equilibrium, (i) the consumer chooses firm A (B) if an only if she’d choose the first (second) offer in the associated main game’s MCU equilibrium and (ii), upon learning their types, the firms choose the same values as they would in the associated MCU equilibrium. Thus, once types are endowed, the MCUSTU equilibrium-path is the same as the equilibrium-path of the corresponding MCU equilibrium. The intuition is that, after the consumer receives her signal, she faces the same kind of decision problem as in the main game and so estimates that A’s offer (B’s offer) is better after a signal in Sˆ1 (in Sˆ2 ) – see Lemma OA3, below. Thus, (i) she follows her signal and (ii) Pr(Sˆ1 |v, v 0 ) (Pr(Sˆ2 |v, v 0 )) is the probability the consumer’s signal indicates A’s (B’s) offer is better. Result (i) ensures that, once the firms’ receive their types, their payoffs are the same as in the reduced game of the associated main game. Thus, the firms behave in the same manner and we obtain equivalence. In light of this, all of our results from the main 9

Symmetry, here, reflects the idea that the consumer’s examination depends only on the offers’ characteristics, not on the identities of the offers’ originators.

15

text continue to hold. For instance, when the signal increases in the more discerning order, Proposition 4 gives that the firms’ values increase whenever they have the same types. Result (ii) ensures that, in the type-uncertainty game, the more discerning order continues to model increases in discernment; see the discussion after Lemma OA3, below. To prove the proposition, we’ll proceed as in the main text. We first need to characterize the consumer’s behavior and show she follows “φU ” (Lemmas OA3 and OA4). Second, we consider an “auxiliary game” between the firms after nature endows types (θA , θB ) and when the consumer follows her signal (see below). We’ll argue that the minimal equilibrium of this game corresponds to the values the firms choose in a MCUSTU equilibrium when their types are (θA , θB ) (Lemma OA5). We’ll then show that the auxiliary game is equivalent to the reduced game of the associated main game (Lemma OA6). The proposition then follows. Lemma OA3. Expected Values. If the firms play symmetric strategies (σA , σB ) and Regularity Condition II holds, then a signal s1 ∈ S1 implies that E(VˆA |σA , σB , s1 ) ≥ E(VˆB |σA , σB , s1 ) and a signal s2 ∈ S2 implies that E(VˆA |σA , σB , s2 ) ≤ E(VˆB |σA , σB , s2 ). Proof. We first establish that that a signal s1 ∈ S1 implies E(VˆA |σA , σB , s1 ) ≥ E(VˆB |σA , σB , s1 ). Since σA (H, H) = σB (H, H), σA (H, L) = σB (L, H), σA (L, H) = σB (H, L), and σA (L, L) = σB (L, L), a bit of algebra gives that E(VˆA |σA , σB , s1 ) ≥ E(VˆB |σA , σB , s1 ) if and only if (σA (H, L) − σB (H, L))(fSˆ (s|σA (H, L), σB (H, L)) − fSˆ (s|σB (H, L), σA (H, L))) ≥ 0. If σA (H, L) = σB (H, L), then the inequality clearly obtains. If σA (H, L) > σB (H, L), strict pointing gives fSˆ (s|σA (H, L), σB (H, L)) > fSˆ (s|σB (H, L), σA (H, L)), so the inequality obtains. Likewise, if σA (H, L) < σB (H, L), strict pointing gives fSˆ (s|σA (H, L), σB (H, L)) < fSˆ (s|σB (H, L), σA (H, L)), so the inequality obtains. Analogous logic allows us to conclude that a signal s2 ∈ S2 implies E(VA |σA , σB , s2 ) ≤ E(VB |σA , σB , s2 ). In light of this result, Pr(Sˆ1 |v, v 0 ) is the probability the signal indicates A’s offer is better and Pr(Sˆ2 |v, v 0 ) is the probability the signal indicates that B’s offer is better. Thus, the more discerning order continues to model increases in discernment. To see this, suppose we increase Pr(Sˆ1 |v, v 0 ) and Pr(Sˆ2 |v 0 , v) for v > v 0 . Then, we also increase the probability the signal indicates A’s offer (B’s offer) is better, conditional on it actually being better, and so increase the probability the signal correctly indicates the better offer.10 If we think of the “better offer” as the better of the relabeled offers (v, v 0 ) after nature has endowed types, the probability S correctly indicates it is Pr(Sˆ1 |v, v 0 ) when v > v 0 , is Pr(Sˆ2 |v, v 0 ) when v < v 0 , and is 1 when v = v 0 . If we think of the “better offer” as the better of the firms’ offers, then probability S correctly indicates the better offer is 21 (1 + Pr(S1 |vA , vB )), where (vA , vB ) = (σA (H, L), σB (H, L)), since the firms 10

16

To digress, the reason our notion of more discernment remains valid, and indeed the reason for Lemma OA3, Proposition 1, and Lemma OA1, is that the joint distribution of the offers’ values is exchangeable: in the main game (and offer game) this occurs because of the second stage and here it occurs because the consumer does not observe the firms’ types. It’s not hard to show that if a Bayesian decision maker (i) has two actions x and y, (ii) an exchangeable prior distribution over the actions’ values, and (iii) receives a signal S˜ about the actions’ values (vx , vy ) that points, then she estimates that x is more valuable after a signal in S˜1 and that y is more valuable after a signal in S˜2 . In general, this result tells us that, so long as the consumer’s (equilibrium) prior about the offers’ values is exchangeable and her signal points, (i) our formulation of more discernment remains economically valid and (ii) the consumer does best by following her signal. It follows that we can replace the second stage of the main game with a number of different modeling devices, including a type-uncertainty game with independently distributed types. Lemma OA4. The Consumer’s Strategy. If the firms play symmetric strategies (σA , σB ) and Regularity Condition II holds, then the consumer’s unique s-undominated strategy is σU , which tells her to choose A after a signal in S1 and B after a signal in S2 , i.e., σU (Sˆ1 ) = A and σU (Sˆ2 ) = B. Proof. It’s immediate from Lemma OA3 that σU is a s-undominated strategy for the consumer. The argument for uniqueness parallels the Proof of Proposition 2. To sketch it, suppose there’s another s-undominated strategy σ 0 . Since both generate the same payoff to the consumer for all symmetric (σA , σB ) and yet differ, it must be that there’s a signal s after which E(VˆA |σA , σB , s) = E(VˆB |σA , σB , s) for all symmetric (σA , σB ). This happens if and only if (σA (H, L) − σB (H, L))(fSˆ (s|σA (H, L), σB (H, L)) − fSˆ (s|σB (H, L), σA (H, L))) = 0. Since we can always choose σA (H, L) 6= σB (H, L), this inequality only obtains if fSˆ (s|v, v 0 ) = fSˆ (s|v 0 , v) for all (v, v 0 ) ∈ Vˆ 2 , which is exactly what strict pointing rules out. It helps to introduce an auxiliary game, which captures the interaction between the firms after nature endows their types (θA , θB ) ∈ Θ2 and when the consumer follows σU . Formally, the auxiliary game beginning (θA , θB ) is game where firms A and B simultaneously pick follow symmetric strategies. In both interpretations, S S 0 implies S 0 correctly indicates the better offer more often than S.

17

vˆA ∈ Vˆ and vˆB ∈ Vˆ and the receive payoffs π ˆAA (ˆ vA , vˆB , θA , θB ) = π ˆA (g(ˆ vA ), g(ˆ vB ), σU , θA , θB ) = wˆθA (ˆ vA ) Pr(Sˆ1 |ˆ vA , vˆB ) vA , vˆB , θA , θB ) = π ˆB (g(ˆ vA ), g(ˆ vB ), σU , θA , θB ) = wˆθ (ˆ π ˆBA (ˆ vB ) Pr(Sˆ2 |ˆ vA , vˆB ), B

where, for each k ∈ R, g(k) denotes a map from Θ2 to R that always takes value k. An equilibrium of this game is a pair (ˆ vA? , vˆB? ) such that πAA (ˆ vA? , vˆB? ) ≥ πAA (ˆ vA , vˆB? ) for all vˆA ∈ VˆA and πBA (ˆ vA? , vˆB? ) ≥ πBA (ˆ vA? , vˆB ) for all vˆB ∈ VˆB . Lemma OA5. A Characterization of MCUSTU equilibria. A strategy vector (σA? , σB? , σC? ) is a MCUSTU equilibrium if and only if (i) σC? = σU and (ii), for each (θA , θB ) ∈ Θ2 , the values (σA? (θA , θB ), σB? (θA , θB )) are the minimal equilibrium of the auxiliary game beginning (θA , θB ). Proof. Tedious, but straightforward. The proof requires four preliminary facts. Fact 0 If (ˆ vA , vˆB ) is the minimal equilibrium of the auxiliary game beginning (θA , θB ), then (ˆ vB , vˆA ) is the minimal equilibrium of the auxiliary game beginning (θB , θA ). Proof. The auxiliary game beginning (θA , θB ) is symmetric to the auxiliary game vB , vˆA , θB , θA ). To see this, write vA , vˆB , θA , θB ) = π ˆBA (ˆ beginning (θB , θA ). That is, π ˆAA (ˆ π ˆAA (ˆ vA , vˆB , θA , θB ) = wˆθA (ˆ vA ) Pr(Sˆ1 |ˆ vA , vˆB ) = wˆθ (ˆ vA ) Pr(Sˆ2 |ˆ vB , vˆA ) A

=

π ˆBA (ˆ vB , vˆA , θB , θA ),

0 ˆ Thus, if B A where the second line follows from the symmetry of S. A (v, θ, θ ) and 0 BA B (v, θ, θ ) denote the firms’ minimal best responses is the auxiliary game beginning A (θ, θ0 ), we have that B A A (v, θA , θB ) = B B (v, θB , θA ). Hence,

vB , θA , θB ) = B A vB , θB , θA ) and vˆB = B A vA , θA , θB ) = B A vA , θB , θA ), vˆA = B A A (ˆ B (ˆ B (ˆ A (ˆ so (ˆ vB , vˆA ) is an equilibrium of the auxiliary game beginning (θB , θA ). All that remains is to prove it’s minimal, we proceed via contradiction. If not, then another equilibrium (ˆ v 0 , vˆ00 ) with either vˆ0 < vˆB or vˆ00 < vˆA . Since an argument analogous to the one we just gave implies (ˆ v 0 , vˆ00 ) is an equilibrium in the auxiliary game beginning (θA , θB ), we have that (ˆ vA , vˆB ) isn’t minimal, a contradiction. Thus, (ˆ vA0 , vˆB0 ) = (ˆ vB , vˆA ). Fact 1 If (σA , σB ) are symmetric strategies such that, for each (θA , θB ) ∈ Θ2 , the pair (σA (θA , θB ), σB (θA , θB )) is an equilibrium of the auxiliary game beginning (θA , θB ), then (σA , σB , σU ) is a symmetric type-uncertainty equilibrium. 18

Proof. We argue via contradiction. If not, then (σA , σB , σU ) isn’t an equilibrium. Thus, one firm, say A, does strictly better by playing σ 0 instead of σA . (The consumer won’t defect by Lemma OA4.) Since all types are equally likely, there is at least one 0 0 0 0 0 0 pair of of types (θA , θB ) ∈ Θ2 such that π ˆA (σ 0 , σB , θA , θB )>π ˆA (σA , σB , θA , θB ). Since 0 0 0 0 0 A 0 0 0 0 0 π ˆA (σ , σB , θA , θB ) = wθA0 (ˆ v ) Pr(Sˆ1 |ˆ v , vˆB ) = π ˆA (ˆ v , vˆB , θA , θB ) and π ˆA (σA , σB , θA , θB )= 0 0 0 0 0 0 ˆ vA , vˆB ) = π wθA0 (ˆ vA ) Pr(S|ˆ ˆAA (ˆ vA , vˆB , θA , θB ), where vˆ0 = σ 0 (θA , θB ), vˆA = σA (θA , θB ) and 0 0 A 0 0 0 A 0 0 0 0 vˆB = σB (θA , θB ), we have π ˆA (ˆ v , vˆB , θA , θB ) > π ˆA (ˆ vA , vˆB , θA , θB ). Thus, (σA (θA , θB ), 0 0 0 0 σB (θA , θB )) isn’t an equilibrium of the auxiliary game beginning (θA , θB ), a contradiction. Fact 2 If (σA , σB ) are strategies such that, for each (θA , θB ) ∈ Θ2 , the pair (σA (θA , θB ), σB (θA , θB )) is the minimal equilibrium of the auxiliary game beginning (θA , θB ), then (σA , σB , σU ) is a symmetric type-uncertainty equilibrium. Proof. Fact 0 implies that (σA , σB ) are symmetric. This is easily seen by contradiction. Suppose (σA , σB ) is not symmetric, then there’s a pair of types (θA , θB ) such that σA (θA , θB ) 6= σB (θB , θA ). Yet, (σA (θA , θB ), σB (θA , θB )) is the minimal equilibrium of the auxiliary game beginning (θA , θB ). Thus, (σB (θA , θB ), σA (θA , θB )) is the minimal equilibrium of the auxiliary game beginning (θB , θA ) by fact zero. Hence, (σA (θB , θA ), σB (θB , θA )) = (σB (θA , θB ), σA (θA , θB )), so σA (θA , θB ) = σB (θB , θA ), a contradiction. The fact now follows immediately from Fact 1. Fact 3 If (σA , σB , σU ) is an equilibrium, then (σA (θA , θB ), σB (θA , θB )) is an equilibrium of the auxiliary game beginning (θA , θB ) for each (θA , θB ) ∈ Θ2 . 0 0 ) such that , θB Proof. We argue via contradiction. If not, then there are types (θA 0 0 0 0 0 0 (σA (θA , θB ), σB (θA , θB )) isn’t an equilibrium of the auxiliary game beginning (θA , θB ). 0 Then one of the firms, say A, has incentive to defect and play v 6= σA (θA , θB ) in said auxiliary game. Yet, because of the construction of the game and the fact that all types are equally likely, A must also do strictly better by playing 0 0 v 0 if (θA , θB ) = (θA , θB ) σ 0 (θA , θB ) = σ (θ , θ ) else A A B in the type-uncertainty game. In symbols, 1 A 0 ˆ (v , σB (θA , θB ), θA , θB ) > 0. π ˆA (σ 0 , σB , σU ) − π ˆA (σA , σB , σU ) = π 4 A Thus, (σA , σB , σU ) isn’t an equilibrium, a contradiction. This concludes the list of facts. With these facts in hand, it’s possible to prove the lemma. 19

We first show that if

(σA? (θA , θB ), σB? (θA , θB )) is the minimal equilibrium of the auxiliary game beginning (θA , θB ) for each (θA , θB ) ∈ Θ2 and if σC? = σU , then (σA? , σB? , σC? ) is a MCUSTU equilibrium. By Fact 2, (σA? , σB? , σC? ) is a symmetric equilibrium where the consumer plays an s-undominated strategy. It only remains to show that (σA? , σB? , σC? ) is minimal among the class of symmetric equilibria where the consumer plays an s-undominated strategy. Suppose it weren’t, then there’s another symmetric equilibrium (σA0 , σB0 , σC0 ) where the consumer plays an sundominated strategy σC0 with σA0 (θA , θB ) < σA? (θA , θB ) or σB0 (θA , θB ) < σB? (θA , θB ) for some type pair (θA , θB ). Since σC0 = σU by Lemma OA4, we have that (σA0 , σB0 , σC0 ) = (σA0 , σB0 , φU ). Thus, Fact 3 gives (σA0 (θA , θB ), σB0 (θA , θB )) is an equilibrium of the auxiliary game beginning (θA , θB ) with σA0 (θA , θB ) < σA? (θA , θB ) or σB0 (θA , θB ) < σB? (θA , θB ), so (σA? (θA , θB ), σB? (θA , θB )) isn’t minimal, a contradiction. We next show that if (σA? , σB? , σC? ) is a MCUSTU equilibrium, then (i) σC? = σU and (ii) (σA? (θA , θB ), σB? (θA , θB )) is the minimal equilibrium of the auxiliary game beginning (θA , θB ) for each (θA , θB ) ∈ Θ2 . Since (i) is obvious in light of Lemma OA4, we only need to prove (ii). Fact 3 gives that (σA? (θA , θB ), σB? (θA , θB )) is an equilibrium of the auxiliary game beginning (θA , θB ) for each (θA , θB ) ∈ Θ2 . It only remains to show minimality in the 0 0 ) such , θB auxiliary game, we argue by contradiction. If not, then there’s a pair of types (θA 0 0 0 0 that (σA? (θA , θB ), σB? (θA , θB )) isn’t the minimal equilibrium of the auxiliary game beginning 0 0 0 0 (θA , θB ), so there’s an equilibrium of this auxiliary game (ˆ vA0 , vˆB0 ) with either vˆA0 < σA? (θA , θB ) 0 0 ). Recall, from the proof of Fact 0, that (ˆ vB0 , vˆA0 ) is an equilibrium of the , θB or vˆB0 < σB? (θA 0 0 ). Let , θA auxiliary game beginning (θB 0 0 vˆ0 if (θA , θB ) = (θA , θB ) A 0 0 0 σA (θA , θB ) = vˆB0 ) , θA if (θA , θB ) = (θB σ ? (θ , θ ) else A A B and

0 0 vˆ0 if (θA , θB ) = (θA , θB ) B 0 0 σB0 (θA , θB ) = vˆA0 if (θA , θB ) = (θB , θA ) σ ? (θ , θ ) else. A A B

Then, (σA0 , σB0 , σU ) is a symmetric strategy vector such that, for each (θA , θB ) ∈ Θ2 , the pair (σA0 (θA , θB ), σB0 (θA , θB )) is an equilibrium of the auxiliary game beginning (θA , θB ). Thus, Fact 1 gives (σA0 , σB0 , σU ) is a symmetric equilibrium where the consumer plays an s-undominated strategy and where σA? 6≤ σA0 or σB? 6≤ σB0 , so (σA? , σB? , σC? ) isn’t a MCUSTU equilibrium, a contradiction.

20

Lemma OA6 The Auxiliary Game Beginning (θA , θB ) and the Reduced Game Let Regularity Condition II hold. For each (θA , θB ) ∈ Θ2 , a pair (ˆ vA? , vˆB? ) is the minimal equilibrium of the auxiliary game beginning (θA , θB ) if and only if (ˆ vA? , vˆB? ) = (vA? (θA , θB ), vB? (θA , θB )), where (vA? (θA , θB ), vB? (θA , θB )) is the unique MCU equilibrium of the associated main game for types (θA , θB ). Proof. In the auxiliary game beginning (θA , θB ), the firms’ payoffs are π ˆAA (ˆ vA , vˆB , θA , θB ) = wˆθA (ˆ vA ) Pr(Sˆ1 |ˆ vA , vˆB ) and π ˆBA (ˆ vA , vˆB , θA , θB ) = wˆθB (ˆ vB ) Pr(Sˆ1 |ˆ vB , vˆA ) (by the symmetry of ˆ and each firm’s set of feasible values is V. ˆ In the associated main game for types (θA , θB ), S) ˆ the firms’ sets of feasible values are VA = VB = V, ˆ and the the consumer’s signal is S, firms’ earnings are wA (v) = wθA (v) and wB (v) = wθB (v). Thus, in the reduced game of this main game, the firms’ payoffs are πAR (vA , vB ) = wˆθA (vA ) Pr(Sˆ1 |vA , vB ) and πBR (vA , vB ) = ˆ and each firms’ set wˆθB (vA ) Pr(Sˆ2 |vA , vB ) = wˆθB (vA ) Pr(Sˆ1 |vB , vA ) (by the symmetry of S) ˆ Hence, the firms have same payoffs and can take the same actions of feasible values is V. in both the auxiliary game beginning (θA , θB ) and the reduced game of the associated main game for types (θA , θB ). It follows that the minimal equilibrium of both games is the same. Thus, Lemma OA6 follows from Lemma 3. Proof of Proposition OA2. If (σA? , σB? , σC? ) is a MCUSTU equilibrium, then part (i) follows from Lemma OA4 and the definition of φU , while part (ii) from Lemmas OA5 and OA6. If (i) and (ii), then Lemmas OA5 and OA6 apply and gives the desired result.

5

Discernment and Other Information Orders

Two classical orders in decision theory are Blackwell’s [1] sufficiency and Lehmann’s [3] accuracy.11 Under suitable assumptions, if two random variables are ordered by sufficiency or accuracy, then the one that’s higher in the order provides a decision maker a higher payoff. Since, all else equal, a more discerning signal also ensures that our consumer gets a higher payoff, it’s natural to wonder if the more discerning order is a weakened form of the sufficiency or accuracy orders. In this section, we present five examples that investigate this question. The first example shows that, for a large and important class of problems, the more discerning order is a weakened form of accuracy and, thus, of sufficiency. The second and third examples show that, outside of this class, the more discerning and accuracy orders are independent. The fourth and fifth examples show that, generally, the more discerning and sufficiency orders are independent. 11

See DeGroot [2] for an overview of statistical decision theory, including Blackwell’s work.

21

Accuracy Lehmann’s accuracy order is only defined for random variables that are functions of unidimensional states and display the monotone likelihood ratio property. Let T and T 0 be two random variables with common support T ⊂ R and conditional cumulative distribution functions FT (t|x) and FT 0 (t|x), where x ∈ R. Then, Lehmann [3] says T 0 is more accurate 0 than T if (i) FT−1 0 (FT (t|x)|x) is nondecreasing in x for every t ∈ T and (ii) T and T have the monotone likelihood ratio property in x,12 where FT−1 0 (y|x) = inf{z ∈ R+ |FT 0 (z|x) ≥ y} is the “generalized inverse” of FT 0 (t|x). Since the consumer’s signal S is two-dimensional, we cannot generally rank signals by accuracy. However, when two signals S and S 0 only depend on the “difference in the offers’ values,” then we can rank them by accuracy. Formally, a signal S depends on the difference in the offers’ values if there’s a family of continuous functions {hs : D → R}s∈S P such that fS (s|v, v 0 ) = hs (v − v 0 ) for each s ∈ S and s∈S hs (x) = 1 for all x ∈ D. Let T be a random variable with support S and conditional mass function fT (s|x) such that fT (s|x) = hs (x) for each x ∈ D, where D = {t ∈ R|t = v − v 0 for some (v, v 0 ) ∈ V 2 }. We call T the associated random variable of S. At the risk of being repetitive, notice that for each (v, v 0 ) ∈ V 2 , both S and T have the same conditional mass function. With this in mind, we say that S 0 is more accurate than S if and only if T 0 is more accurate than T , where T 0 is the associated random variable of S 0 . The next example presents a class of signals where more accurate signals are always more discerning. This class is important because it encompasses all latent variable signals with mean-zero and symmetric noise – e.g., Examples 1 to 4, as well as Examples OA1 to OA4. The subsequent two examples show that outside of this class, accuracy and discernment are independent orders. Example OA5. Accuracy and Sufficiency Imply Discernment. Consider two binary signals S and S 0 that depend on the differences in the offers’ values. Both have common support S = {1, 2} and conditional mass functions g (v 0 − v) g 0 (v 0 − v) if s = 1 if s = 1 S S fS (s|v, v 0 ) = and fS 0 (s|v, v 0 ) = 1 − g (v 0 − v) if s = 2 1 − g 0 (v 0 − v) if s = 2, S S where gS : R → (0, 1) and gS 0 : R → (0, 1) are strictly decreasing and continuous functions with gS (0) = gS 0 (0) = 12 .13 Clearly, S has associated random variable T with fT (1|x) = gS (x) 12 When T has a density (or probability mass function) fT (t|x) and T 0 has a density fT 0 (t|x), to say that say that T and T 0 have monotone likelihood ratio property in x is to mean that the ratios fT (t|x0 )/fT (t|x) and fT 0 (t|x0 )/fT 0 (t|x) are increasing in t for any x < x0 ; see Lehmann [3] for details. 13 The signals in Examples 1 and 2 are of this form. Recall, for instance, that in Example 1, fS (1|v, v 0 ) =

22

and fT (2|x) = 1 − gS (x), while S 0 has associated random variable T 0 with fT 0 (1|x) = gS 0 (x) and fT 0 (2|x) = 1 − gS 0 (x). It’s easily seen that both signals strictly point, with S1 = S10 = {1} and S2 = S20 = {2}, and that both T and T 0 have the monotone likelihood property in x because gS (x) and gS 0 (x) are strictly decreasing. Thus, we can rank both in the more discerning order and in the accuracy order. It’s readily seen that S 0 is more discerning than S if gS (x) ≤ gS 0 (x) when x ≤ 0 and gS (x) ≥ gS 0 (x) when x ≥ 0. The signal S 0 is more accurate than S if FT−1 0 (FT (s|x)|x) is nondecreasing in x for every s ∈ S, where FT (x|x) is the cumulative distribution function of T and 0 if y = 0 FT−1 0 (y|z) = inf{x ∈ R+ |FT 0 (x|z) ≥ y} = 1 if 0 < y ≤ gS 0 (z) 2 if g 0 (z) < y ≤ 1, S

is the generalized inverse of the cumulative distribution function of T 0 . Claim OA2. If S 0 is more accurate than S, then S 0 is more discerning than S. Proof. Since FT−1 0 (FT (2|x)|x) = 2 is nondecreasing in x, we only need concern ourselves with −1 FT 0 (FT (1|x)|x). We have 0 if gS (x) = 0 −1 FT 0 (FT (1|x)|x) = 1 if gS (x) − gS 0 (x) ≤ 0 2 if g (x) − g 0 (x) > 0. S S −1 (Since gS (x) > 0, we can ignore the case of FT−1 0 (FT (1|x)|x) = 0.) Since FT 0 (FT (1|x)|x) is nondecreasing in x, once its value increases from 1 to 2 it never reduces. Thus, there’s a single point xˆ such that (i) gS (x) − gS 0 (x) ≤ 0 for all x ≤ xˆ and (ii) gS (x) − gS 0 (x) > 0 for all x > xˆ. Since gS (0) = gS 0 (0) = 1/2, we must have xˆ = 0. Hence, gS (x) ≤ gS 0 (x) when x ≤ 0 and gS (x) ≥ gS 0 (x) when x ≥ 0, so S 0 is more discerning than S.

Since Blackwell’s sufficiency order implies Lehmann’s accuracy order (see Lehmann [3]), it follows that our more discerning order is a weaker form of both orders, at least for the class of signals fitting this example. 4 Example OA6. Discernment Does Not Imply Accuracy. 0

0

0

v−v v −v 0 0 Φ( v−v σ ). Since Φ(·) is symmetric, Φ( σ ) = 1 − Φ( σ ). Thus, fS (1|v, v ) = gS (v − v) with gS (x) = x 1 − Φ( σ ), which is easily seen to meet all of the above regularity conditions. The argument for Example 2 is similar.

23

Consider a variant of Example C1, where S and S 0 have conditional mass functions g 0 (v − v 0 ) g (v − v 0 ) if s = 1 if s = 1 S S and fS 0 (s|v, v 0 ) = fS (s|v, v 0 ) = 1 − g 0 (v − v 0 ) if s = 2, 1 − g (v − v 0 ) if s = 2 S S where gS : R → (0, 1) and gS 0 : R → (0, 1) are strictly increasing functions with gS (0) = gS 0 (0) = 21 . Clearly, S has associated random variable T with fT (1|x) = gS (x) and fT (2|x) = 1 − gS (x), and S 0 has associated random variable T 0 with fT 0 (1|x) = gS 0 (x) and fT 0 (2|x) = 1 − gS 0 (x). It’s easily seen that both signals strictly point with S1 = S10 = {1} and S2 = S20 = {2}. In addition, it’s readily verified that S 0 is more discerning than S when gS (x) ≥ gS 0 (x) when x ≤ 0 and gS (x) ≤ gS 0 (x) when x ≥ 0. However, T and T 0 fail to have the monotone likelihood ratio property in x because gS (x) and gS 0 (x) aren’t decreasing. Thus, S and S 0 are unranked in the accuracy order. Example OA7. Accuracy Does Not Imply Discernment. Consider a variant of Example C1, where the only change we make is that gS (1) = gS 0 (1) = 21 and gS (x) 6= gS 0 (x) for any x 6= 1. It’s easily verified that the two signals still strictly point, with S1 = S10 = {1} and S2 = S20 = {2}, and that the associated random variables have the monotone likelihood ratio property in x. Familiar logic applies and gives that if S 0 is more accurate than S, then (i) gS (x) < gS 0 (x) when x < 1 and (ii) gS (x) > gS 0 (x) when x > 1. Requirement (i) is actually incompatible with pointing. Simply, for v ≥ v 0 , fS 0 (2|v 0 , v) ≥ fS (2|v 0 , v) holds if and only if gS (v − v 0 ) ≥ gS 0 (v − v 0 ). But, when 0 ≤ v − v 0 < 1, (i) gives that this cannot be the case. Thus, accuracy doesn’t imply discernment. 4 Sufficiency Blackwell’s sufficiency order is defined for all random variables (see Blackwell [1] and DeGroot [2]). However, for simplicity, we focus on binary signals with support S = {1, 2}. Applying his definition (as generalized by DeGroot [2]) to our binary setting gives that S 0 is more sufficient than (alternatively, sufficient for) S if there are nonnegative weights b11 , b12 , b21 , and b22 such that fS (1|v, v 0 ) = b11 fS 0 (1|v, v 0 ) + b12 fS 0 (2|v, v 0 ), fS (2|v, v 0 ) = b21 fS 0 (1|v, v 0 ) + b22 fS 0 (2|v, v 0 ) b11 + b21 = 1, and b12 + b22 = 1. The next two examples show that the discerning order and the sufficiency order are inde-

24

pendent.14 Example OA8. Discernment Does Not Imply Sufficiency. Consider two binary signals S and S 0 , with common support S = {1, 2} and conditional mass functions 1 if s = 1, v > v 0 , & |v − v 0 | ≤ 1 3 1 2 0 0 if s = 2, v > v , & |v − v | ≤ 1 if s = 1 & v > v 0 3 3 1 2 0 0 if s = 1, v > v , & |v − v | > 1 if s = 2 & v > v 0 6 3 0 1 fS (s|v, v 0 ) = 65 if s = 2, v > v 0 , & |v − v 0 | > 1 & fS 0 (s|v, v ) = 100 if s = 1 & v < v 0 99 if s = 2 & v < v 0 1 if s = 1 & v < v 0 100 100 99 1 0 if s = 2 & v < v if v = v 0 . 100 2 1 if v = v 0 . 2

It’s easily seen that both signals strictly point, with S1 = S10 = {1} and S2 = S20 = {2}, and that S 0 is more discerning than S. Nevertheless, S and S 0 are unordered in the sense of sufficiency. This follows from the fact that the distribution of S depends on the distance between v and v 0 , whereas S 0 does not. We now illustrate that S 0 is not sufficient for S via contradiction; an analogous argument gives that S is not sufficient for S 0 . If S 0 were sufficient for S, then there are nonnegative weights b11 , b12 , b21 , and b22 such that fS (1|v, v 0 ) = b11 fS 0 (1|v, v 0 ) + b12 fS 0 (2|v, v 0 ), fS (2|v, v 0 ) = b21 fS 0 (1|v, v 0 ) + b22 fS 0 (2|v, v 0 ) b11 + b21 = 1, and b12 + b22 = 1. Suppose there were such weights. Then, at any v > v 0 , we have fS 0 (1|v, v 0 ) = fS 0 (2|v, v 0 ) = 23 , so b11 fS 0 (1|v, v 0 ) + b12 fS 0 (2|v, v 0 ) is a point in R. Yet, fS (1|v, v 0 ) =

1

if |v − v 0 | ≤ 1

1

if |v − v 0 | > 1.

3 6

1 3

and

Thus, fS (1|v, v 0 ) 6= b11 fS 0 (1|v, v 0 ) + b12 fS 0 (2|v, v 0 ). 4 Example OA9. Sufficiency Does Not Imply Discernment. Consider two binary signals S and S 0 , with common support S = {1, 2} and conditional 14

These examples have discontinuous conditional mass functions for simplicity. Examples satisfying continuity can be constructed, but are unwieldy.

25

mass functions

fS (s|v, v 0 ) =

4 9 5 9

101 300

199 300 1 2

if s = 1 & v > v

0

if s = 2 & v > v 0 0 if s = 1 & v < v 0 and fS 0 (s|v, v ) =

if s = 2 & v < v 0 if v = v 0

1 3 2 3

1

100 99 100 1 2

if s = 1 & v > v 0 if s = 2 & v > v 0 if s = 1 & v < v 0 if s = 2 & v < v 0 if v = v 0 .

It’s easily seen that both signals strictly point, with S1 = S10 = {1} and S2 = S20 = {2}. However, neither signal is more discerning than the other: S 0 cannot be more discerning because fS 0 (1|v, v 0 ) = 13 6≥ fS (1|v, v 0 ) = 49 when v > v 0 , and S cannot be more discerning 99 6≤ fS (2|v 0 , v) = 199 when v > v 0 . Nevertheless, the signals are because fS 0 (2|v 0 , v) = 100 300 ordered by sufficiency; specifically, S 0 is sufficient for S. That is, there are nonnegative weights b11 = 32 , b12 = 31 , b21 = 13 , and b22 = 23 such that fS (1|v, v 0 ) = b11 fS 0 (1|v, v 0 ) + b12 fS 0 (2|v, v 0 ), fS (2|v, v 0 ) = b21 fS 0 (1|v, v 0 ) + b22 fS 0 (2|v, v 0 ) b11 + b21 = 1, and b12 + b22 = 1. We omit the verification. 4

6

Strong More Discernment without Symmetry

In the main text, we only developed the strongly more discerning order for symmetric signals. In this section, we generalize the order to asymmetric signals and show that Propositions 5 to 7 continue to hold. If a pointing signal S is asymmetric, then statements about the properties of PrS (S1 |v, v 0 ) do pin down the properties of PrS (S2 |v 0 , v). With this in mind, we extend the definition of strongly more discerning as follows. Definition. For signals S and S 0 that strictly point, we say S 0 is strongly more discerning ˚G S 0 , if (PrS 0 (S10 |v, v 0 ) + PrS 0 (S20 |v 0 , v))/(PrS (S1 |v, v 0 ) + PrS (S2 |v 0 , v)) is a than S, denoted S weakly increasing function of v on V for each v 0 ∈ V. For symmetric signals S and S 0 , this definition is clearly equivalent to the one given in the ˚G S 0 , then we also implicitly assume that S and S 0 main text. Note that if we assume S strictly point. ˚G S 0 need not imply S S 0 . The intuition lies in the nature of the Unfortunately, S 26

orders. To fix ideas, suppose the firms choose values (vA , vB ) with vA ≥ vB . The more discerning order ensures that, regardless of whether nature assigns firm A the first or second slot, that the consumer has a higher chance of selecting A’s more valuable offer, i.e., that PrS 0 (S10 |vA , vB ) ≥ PrS (S1 |vA , vB ) and that PrS 0 (S20 |vB , vA ) ≥ PrS (S2 |vB , vA ). In contrast, the strongly more discerning order only ensures that:15 1 1 (PrS 0 (S10 |vA , vB ) + PrS 0 (S20 |vB , vA )) ≥ (PrS (S1 |vA , vB ) + PrS (S2 |vB , vA )). 2 2

(3)

That is, on average before nature determines the firms’ order, the consumer has a higher chance of selecting A’s better offer. While the more discerning order also ensures this, equation (3) does not imply that the consumer has a higher chance of selecting A’s better offer for each of nature’s possible relabelings absent symmetry – in symbols, equation (3) does not imply PrS 0 (S10 |vA , vB ) ≥ PrS (S1 |vA , vB ) and PrS 0 (S20 |vB , vA ) ≥ PrS (S2 |vB , vA ) absent symmetry. Hence, the strongly more discerning order can be independent of the more discerning order. We thus omit asymmetric signals from the main text in order to keep the discussion focused on the effects of more discernment. The generalized notion of strong more discernment still reflects the idea that the consumer is better at recognizing change in the offer’s characteristics and values. However, it’s now in an average sense across nature’s possible orderings of the firms’ offers as opposed to the pointwise sense of main text, where it holds for each of nature’s possible orderings. Critically, the strongly more discerning order still implies that there’s a complementarity between signals ranked in the order and each firm’s value in the reduced game. To see this, ˚G S 0 , then suppose the firms choose values (vA , vB ) in the reduced game. If S 0

PrS 0 (S10 |vA , vB ) + PrS 0 (S20 |vB , vA ) mSA (vA , vB ) = PrS (S1 |vA , vB ) + PrS (S2 |vB , vA ) mSA (vA , vB ) is weakly increasing in vA and 0

mSB (vA , vB ) PrS 0 (S10 |vB , vA ) + PrS 0 (S20 |vA , vB ) = PrS (S1 |vB , vA ) + PrS (S2 |vA , vB ) mSB (vA , vB ) is weakly increasing in vB . Hence, if either A or B unilaterally increases it’s value to vA0 or 15

When v ≥ v 0 , we have PrS 0 (S10 |v, v 0 ) + PrS 0 (S20 |v 0 , v) PrS 0 (S10 |v 0 , v 0 ) + PrS 0 (S20 |v 0 , v 0 ) ≥ = 1. PrS (S1 |v, v 0 ) + PrS (S2 |v 0 , v) PrS (S1 |v 0 , v 0 ) + PrS (S2 |v 0 , v 0 )

It follows that 1/2(PrS 0 (S10 |v, v 0 ) + PrS 0 (S20 |v 0 , v)) ≥ 1/2(PrS (S1 |v, v 0 ) + PrS (S2 |v 0 , v)).

27

vB0 , we have

0

0

mSA (vA , vB ) mSB (vA , vB0 ) mSB (vA , vB0 ) mSA (vA0 , vB ) ≥ or ≥ 0 0 mSA (vA , vB ) mSB (vA , vB ) mSA (vA , vB ) mSB (vA , vB )

and so obtain the same kind of complementarity as in the main text. Hence, if it’s a good idea for either firm to increase their offer’s value when the consumer’s signal is S, then it’s also a good idea for the firm to do so when the consumer’s signal is S 0 . Consequently, the firms’ reduced game best responses shift up as the signal increases and standard methods yield the following propositions. Proposition OA3. Analogue of Proposition 5. ˚G S 0 , then the firms offer greater If S and S 0 are log-supermodular signals such that S values in the MCU equilibrium when the consumer’s signal is S 0 than when it’s S, i.e., (vA? (S), vB? (S)) ≤ (vA? (S 0 ), vB? (S 0 )). Proof. Analogous to the Proofs of Lemmas 9 and 12, as well as Proposition 5, and therefore omitted. Proposition OA4. Analogue of Proposition 6. Let (i) let S and S 0 be log-supermodular and monotone signals, with the difference property, ˚G S 0 and let (ii) the antecedents of Lemma 11 hold. Then, if (iii) wA (v) is such that S concave and the amounts by which firm A outbids firm B under both signals satisfy B R A (z, S)− R R 0 0 z ≤ BR A (z, S ) − z, where z = B B (v A , S) and z = B B (v A , S ), we have: (a) Firm A offers a higher value than firm B in the MCU equilibrium when the consumer’s signal is S or S 0 , i.e., (vA? (S), vA? (S 0 )) ≥ (vB? (S), vB? (S 0 )) (b) The consumer’s chance of selecting A’s better offer is higher in the MCU equilibrium 0 0 when her signal is S 0 instead of S, i.e., mSA (vA? (S 0 ), vB? (S 0 )) ≥ mSA (vA? (S 0 ), vB? (S 0 )). (c) The consumer’s payoff is higher when her signal is S 0 instead of S, i.e., π ˜C (S) ≤ π ˜C (S 0 ). (Analogous results obtains if we reverse the roles of the firms.) Proof. Analogous to the Proofs of Lemma 11, Lemma A1 and Proposition 6, and thus omitted. Proposition OA5. Analogue of Proposition 7. Let S and S 0 be log-supermodular, symmetric, and strictly monotone signals with S S 0 . If wB (v) is weakly increasing and PrS 0 (S10 |v, vB? (S)) + PrS 0 (S20 |vB? (S), v) PrS 0 (S10 |vA? (S), vB? (S)) + PrS 0 (S20 |vB? (S), vA? (S)) ≤ PrS (S1 |v, vB? (S)) + PrS (S2 |vB? (S), v) PrS (S1 |vA? (S), vB? (S)) + PrS (S2 |vB? (S), vA? (S)) for all v ≥ vA? (S), then both firms offer weakly lower values in the MCU equilibrium when the consumer’s signal is S 0 than when it’s S, i.e., (vA? (S 0 ), vB? (S 0 )) ≤ (vA? (S), vB? (S)). (An 28

analogous result obtains if we reverse the roles of the firms.) Proof. Analogous to the Proof of Proposition 7 and omitted.

7

More Discernment and Less Competitive Offers

In the main text, we observed out that more discernment rotates the firms’ best responses and that these rotations must occur in the right ways for one or both of the offers’ values to deteriorate. While it’s generally non-trivial to control these rotations, in this section we focus on a special case of our game and develop a joint condition on the signals and firms’ earnings that ensures more discernment leads to a worse offer by one firm. Proposition OA6. More Discernment and a Worsening of Firm A’s Offer. Let S and S 0 be log-supermodular and symmetric signals with S S 0 , let PrS (S1 |v, v 0 ), PrS 0 (S10 |v, v 0 ), and wA (v) be continuously differentiable and strictly log-concave in v on V, let wA (v A ) = wA (v A ) = 0 and wA (v) > 0 for some v ∈ int(VA ), and let v˜ denote the solution to ∂ ln(PrS (S1 |v, v 0 ) ∂ ln(PrS 0 (S10 |v, v 0 ) = 0 0 ∂v ∂v (v,v )=(v ? (S),˜ v) (v,v )=(v ? (S),v ? (S)) A

A

B

if this equation has a solution on VB , otherwise set v˜ = v B . If (i) ρ(v, vB? (S)) is weakly decreasing in v for all v ≥ vA? (S) and if either: (ii.a) We have v B ≤ v˜. (ii.b) There is a k ∈ VB such that wB (v) = 0 for all v ≥ k and k ≤ v˜. Then vA? (S 0 ) ≤ vA? (S). (An analogous result obtains if we reverse the firms’ roles.) That is, under strong regularity conditions, more discernment leads to worse offers under conditions (i) and (ii.a) or (ii.b). While it’s easy to interpret condition (i) – see the discussion after Lemma 12 in the main text – conditions (ii.a) and (ii.b) are more technical in nature and require knowledge of the MCU equilibrium under S. We thus think of the proposition as giving a test which allows us to determine if A’s offer worsens as the signal shifts, without having to compute the equilibrium under S 0 . The next examples illustrate. Example OA10. Illustration of Proposition OA5 via Condition (ii.a). Let VA = [0, 1], VB = [0, 1/2], wA (v) = v − v 2 , and wB (v) = v − 2v 2 . Consider two signals for the consumer S and S 0 , which are from the family considered in Example 1 and differ in discernment. Specifically, S has noise with standard deviation σS = 1/10 and S 0 has noise with standard deviation σS = 1/50.

29

It’s readily verified that this example meets all of the regularity conditions of the proposition, so we only need to verify conditions (i) and (ii.a) or (ii.b); we’ll see that (ii.a) holds here. Computation gives that the MCU equilibrium values under S are (vA? (S), vB? (S)) = (0.623, 0.457). To see that (i) holds, note that ρ(v, vB? (S)) = Φ(10(v − vB? (S)))/Φ(50(v − vB? (S))). Since Φ(10t)/Φ(50t) is decreasing on [0.1, 2] and since vA? (S) − vB? (S) = 0.166, we have ρ(v, vB? (S)) is decreasing in v for all v ≥ vA? (S). As to (ii.a), we have Φ0 (10(vA? (S) − vB? (S))) ∂ ln(PrS (S1 |vA? (S), vB? (S)) = ∂v Φ(10(vA? (S) − vB? (S))) ∂ ln(PrS 0 (S10 |vA? (S), v) Φ0 (50(vA? (S) − v)) = . ∂v Φ(50(vA? (S) − v)) Solving for v gives v = 0.575, which is greater than v B = 1/2; so we set v˜ = 1/2. Thus, condition (ii.a) holds. Proposition OA5 then gives that A’s offer worsens. Computation verifies this. Specifically it shows that the MCU equilibrium under S 0 is (vA? (S 0 ), vB? (S 0 ), φU ) = (0.550, 0.494, φU ). 4 Example OA11. Another Illustration of Proposition OA5 via Condition (ii.b). Let VA = [0, 1], VB = [0, 1], wA (v) = 1 − v, and wB (v) = max{1 − 2v, 0}. Let S and 0 S be two signals drawn from the same family of distributions in Example 2, with βS = 1/25 and βS 0 = 1/100. It’s readily verified that this example meets all of the regularity conditions of the proposition. It’s readily verified that this example meets all of the regularity conditions of the proposition, so we only need to verify conditions (i) and (ii.a) or (ii.b); we’ll see that (ii.b) holds here. Computation gives that the MCU equilibrium under S is (vA? (S), vB? (S), φU ) = ? (0.549, 0.456, φU ). To see that (i) holds, note that ρ(v, vB? (S)) = (1 + e−25(v−vB (S)) )/(1 + ? e−100(v−vB (S)) ). Since (1+e−25t )/(1+e−100t ) is decreasing on [0.03, 1] and since vA? (S)−vB? (S) = 0.093, we have ρ(v, vB? (S)) is decreasing in v for all v ≥ vA? (S). As to (ii.b), we have ?

?

∂ ln(PrS (S1 |vA? (S), vB? (S)) 25e−25(vA (S)−vB (S)) = ? ? ∂v 1 + 25e−25(vA (S)−vB (S)) ? ∂ ln(PrS 0 (S10 |vA? (S), v) 100e−100(vA (S)−v) = . ? ∂v 1 + 25e−100(vA (S)−v) Solving gives v = 0.511, so we take v˜ = 0.511. Since wB (v) = 0 for all points greater than 0.5, we have k = 0.5, condition (ii.b) thus holds. Hence, Proposition OA5 gives that A’s offer worsens. Computation verifies this. Specifically it shows that the MCU equilibrium under S 0 is (vA? (S 0 ), vB? (S 0 ), φU ) = (0.528, 0.490, φU ). 4 The intuition for the proposition is v˜ is the largest value firm B could play before firm 30

A would play a value above vA? (S); we establish this fact in the proof. (Thus, the value of v˜ is determined by the rotation of A’s best response.) In general, whether B plays a value below or above v˜ depends on the rotation of its best response. The proposition, however, side-steps this issue by making assumptions that ensure B always plays a value below v˜ when the consumer’s signal is S 0 ; this is where conditions (ii.a) and (ii.b) enter. Thus, A’s value falls. Proof of Proposition OA6. Let v˜ be the maximum amount value that B can play before A plays a value above vA? (S) when the consumer’s signal is S 0 . That is, v˜ ∈ VB ? ? v, S 0) = B R solves B R A (˜ A (vB , S) = vA (S) (per Lemmas 3 and 5) since A’s best response is increasing (per Lemma 1), provided this equation has a solution. If this equation has no 0 ? solution, then it must be that B R A (v, S ) < vA (S) for all v ∈ VB because condition (i) ensures R ? ? 0 BR A (vB (S), S ) ≤ B A (vB (S), S) and A’s best response is increasing. Thus, it’s without loss to take v˜ = v B . 0 Conditions (ii.a) or (ii.b) then ensure that B R ˜ for all v ∈ VB , so A’s value B (v, S ) ≤ v 0 falls as the signal shifts. If (ii.a) holds, we automatically have that B R ˜ since B B (v, S ) ≤ v cannot play a value above v B . If (ii.b) holds, all values above k are weakly dominated for B – it’s payoff to such a value is zero, while it’s payoff to any value below k is weakly positive. 0 Thus, B’s smallest best response is no greater than k, i.e., B R B (v, S ) ≤ k, which implies BBR (v, S 0 ) ≤ v˜. It remains to solve for v˜ in terms of the primitives. This is where our regularity conditions v , S 0 ) = BAR (vB? , S). come into play. Assume, for the moment that there is a solution to BAR (˜ It’s readily verified that the regularity conditions ensure that A’s payoff is strictly log-concave and continuously differentiable, and ensure that its best response is interior. Thus, the logfirst order condition holds under both signals and completely describes the unique best response; so, ∂ ln(PrS (S1 |vA? (S), vB? (S)) ∂ ln(wA (vA? (S))) + = 0. ∂vA ∂vA ∂ ln(PrS (S1 |vA? (S), v˜) ∂ ln(wA (vA? (S))) + = 0. ∂vA ∂vA Putting these two equations together gives that v˜ solves ∂ ln(PrS 0 (S10 |vA? (S), v˜) ∂ ln(PrS (S1 |vA? (S), vB? (S)) = . ∂vA ∂vA If this system has no solutions on VB , then BAR (˜ v , S 0 ) = BAR (vB? , S) has no solutions on VB , so we take v˜ = v B per above. 31

References [1] D. Blackwell. Comparison of Experiments. Proceedings of Second Berkeley Symposium Mathematical Statistics and Probability, pages 93–102, 1951. [2] M. DeGroot. Optimal Statistical Decisions. John Wiley & Sons, 1970. [3] E. Lehmann. Comparing Location Experiments. The Annals of Statistics, 16(2):521–533, 1988. [4] M. Nermuth. Information Structures in Economics, volume 196 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, 1982. [5] J. Perloff and S. Salop. Firm-Specific Information, Product Differentiation, and Industry Equilibrium. Oxford Economic Papers, 38:184–202, November 1986.

32