Resultados de encuestas

1 : Suppose is an RE and de…ne Bi : R+ Htd!R by

Bi( i;t; yi;tjhtw) =E (1 ) i;t+ Uijhtd; i;t; yi;t : (6) Consider an on-path history ht

0. Then jht0 is a Perfect Bayesian Equilibrium of the con- tinuation game. In particular, for each on-pathDt; t; dt, and t, agent iis willing to choose

ei;t only if

ei;t 2argmax ei2R+

E (1 ) i;t+ Uijhtd; i;t; ei;t =ei (1 )c(ei): (7) We can rewrite (7) as

ei;t 2argmax ei2R+

E E (1 ) i;t + Uijhtd; i;t; ei;t =ei; yi;t jhtd; i;t; ei;t =ei (1 )c(ei):

Conditional on yi;t,ei;t does not a¤ect continuation play. So this constraint implies(2): Suppose Bi i;t; yi;tjhtd < E Ui h0t+1 htd; i;t; yi;t at some on-path historyhty. Then agent i may pro…tably deviate by choosing i;t = 0 and earning no less than Ui ht0+1 in the continuation game. So the left-hand side of (3) must hold in any RE. Suppose that the principal refuses to pay i;t 0 following hty. Following this deviation, the re…nement to RE has no bite. We claim the principal’s continuation payo¤ after this deviation is bounded from below by E " ; ht₀+1 1 X t0₌₀

(1 ) t0(yi;t+t0 wi;t+t0 i;t+t0) ht_y

#

: (8)

To prove this claim, consider the following strategy for the principal following a deviation in i;t. Agent i observes this deviation, but no other agents do. Denote all variables that are observed by at least one agent j ₆=i by _[j6=i j ht

0 0 . In each period ht 0 0 2 Ht 0 0, the principal plays according to _{j [}j6=i i ht 0

0 , with the sole exception that wi;t0 = i;t0 = 0 for alli2I. This strategy is identical to except for transfer payments. Transfer payments do not a¤ect the continuation game, so this strategy is feasible. Moreover, this strategy and are indistinguishable for every agent j ₆=i. The principal’s payo¤ from this strategy equals (8), so this strategy bounds the principal’s payo¤ from below.

The principal is willing to pay i;t 0 only if

(1 )E i;tjhty E

" ₁ X

t0=1

(1 ) t0(yi;t+t0 wi;t+t0 i;t+t0) ht_y

#

: (9)

Adding Ui to both sides of this expression yields

E (1 ) i;t+ Ui( ; ht0+1)jh t

y E Si( ; ht0+1)jh t y :

This inequality must hold a fortiori in expectation. Because ht_d; _i;t, and yi;t are elements of

ht

y, iterating expectations and applying the de…nition of Bi yields the right-hand inequality of (3).

2: We construct a RE from . Recursively de…ne as follows:

1. Ift= 0, thenht;₀ =ht₀ =_;, the unique null history. Otherwise, begin withht₀; ht;_{0 2 H}t₀

such that (i) ht

0 is on-path for , (ii) h t;

0 is on-path for , and (iii) ht0 and h t; 0 induce identical continuation games. De…ne _jht;₀ as follows, where starred variables represent actions played in and unstarred variables represent actions played in .

(a) Following the realization of t and Dt, the principal chooses history hte 2 Het using distribution _{j f}ht

0; t; Dtg. The principal chooses dt as in hte. For each agenti_{2 f}1; : : : ; N_g, the principal pays

w_i;t =E yi;t 1

1 Bi i;t; yi;tjh t

d Si ; ht0+1 htd; i;t . Notew_i;t 0by (3). The principal sends a message to agent i:

m_i;t =nht;₀ ; ai;t; ei;t; Bi i;t; yi;tjhtd E Si ; ht0+1 jh t

d; i;t; yi;t _y i;t 0

o

:

(b) Agent i chooses ai;t; ei;t as in mi;t.

(c) If output is y_t, then for each agenti_{2 f}1; : : : ; N_g, (1 ) _i;t =Bi i;t; yi;tjh

t

d E Si ; ht0+1 jh t

d; i;t; yi;t

Note that _i;t 0 by (3). Given m_i;t and the realized y_i;t, agent i can perfectly infer _i;t.

(d) Letht₀+1; be the realized history at the end of periodt. The principal drawsht₀+1 ₂

Ht0+1 from j fhte; ytg. Continuation play jht0+1; is constructed by repeating steps (a)-(d) usinght₀+1 and ht₀+1; .

2. If agent i observes a deviation, then he takes his outside option and pays no transfers in this and every subsequent period. If the principal observes a deviation, thenmj;t0 =

wj;t0 = _j;t0 = 0for each agentj 2 f1; : : : ; Ngin each future period. If agentideviates, the principal chooses dt to min-max agent i. Otherwise, dt is chosen uniformly at random following a deviation.

First, note that past e¤ort choices do not a¤ect current play. Moreover, agents know the true history at the start of a period ht;₀ whenever they take actions in that period. Therefore, if we can show that no player has a pro…table deviation from , it immediately follows that _jht;₀ is a PBE for every on-pathht;₀ and hence is a RE. Furthermore, both total continuation surplus andi-dyad surplus for every i_{2 f}1; :::; N_g are identical in _jht;₀

and _jht₀ by construction.

First, consider the principal. For any on-path ht;_d and each agent i _{2 f}1; : : : ; N_g, the distribution overy_i;t is the same as in _jht

d. So

E y_i;t w_{i;t i;t} ht;_d = 0

and ; ht;_d = 0. If the principal deviates in d_t, w_i;t, or m_i;t, then each agent i either knows that this action is a deviation or not. If agent i knows these actions are a deviation, then the principal earns0from agent ibecause that agent rejects production in this and all future periods. If agent i does not know these actions are a deviation, then the principal must announce some on-path history ~ht_d such that i(~htd) = i(h

t;

d ). But then E [yi;t0

w_i;t0 i;t0j~htd] = 0 in every t0 t because h~td is on-path. So the principal cannot pro…tably deviate ind_t; w_i;t, orm_i;t. The principal likewise has no pro…table deviation from _i;t because

i;t 0. So the principal has no pro…table deviation on the equilibrium path.

Suppose the principal deviates o¤ the equilibrium path. As with the argument in the previous period, if this deviation is detected by agent i then the principal earns 0 from agent i. If this deviation is not detected by agent i, then it must be in d_t; w_i;t, or m_i;t, since _i;t >0is always detected as a deviation. But the principal earns payo¤0 following a deviation d_t; w_i;t, orm_i;t by the argument above. So she has no pro…table deviation o¤ the equilibrium path.

Consider agent i. At each on-path history ht;₀ , E ui;tjht;0 = E yi;t c(ei;t)jht;0 : SoUi ; ht;0 =Si ; ht;0 . By construction of ,Si ; ht;0 =Si( ; ht0). Sincewi;t 0, agent i has no pro…table deviation from w_i;t. After agent i observes an on-path w_i;t; m_i;t

and chooses e_i;t, he earns

E (1 ) i;t+ Si( ; ht0+1; ) h t;

d ; wi;t; mi;t; ei;t c(ei;t);

since he can perfectly infer ht;_d from D_t; _t; d_t; and m_i;t. Plugging in the de…nition of i;t yields E hBi i;t; yi;tjh t d E Si ; ht0+1 jh t d; wi;t; yi;t + Si( ;~ht0+1) h t;

d ; wi;t; mi;t; ei;t

i

c(e_i;t):

Now, E Bi( i;t; yi;tjhtd)jh t;

d ; wi;t; mi;t; ei;t = E Bi( i;t; yi;tjhtd)jhtd; ei;t because t; dt are the same inht

dandh t;

d . Similarly,E Si( ; ht0+1)jhtd; i;t; yi;t =E Si( ; ht0+1; )jh t;

d ; wi;t; mi;t; yi;t by construction. Therefore, the agent is willing to choose e_i;t so long as

e_{i;t 2}arg max ei

E Bi i;t; yi;tjhtd jhtd; ei c(ei):

E¤ort e_i;t satis…es this constraint because (2) holds. O¤ the equilibrium path, continuation play is independent of ei;t and so the agent optimally chooses ai;t = 0.

Following any deviation in i;t <0, agent iearns continuation surplus Ui h t+1;

0 . Agent

i observes or infersht;_d ; w_i;t; m_i;t; y_i;t in . So agent iis willing to pay _i;t <0if (1 ) _i;t E Ui ; h0t+1; Ui ht0+1; h

t;

d ; wi;t; mi;t; yi;t Recall that Ui( ; ht0+1; ) =Si( ; ht0+1; ) by construction. Moreover,

E Si ; ht0+1; jh t;

d ; wi;t; mi;t; yi;t =E Si( ; ht0+1)jh t

d; i;t; yi;t

because continuation dyad-surplus in following ht;_d ; w_i;t; m_i;t; y_i;t is drawn is drawn ac- cording to _j(ht

e; yt). Furthermore, E Ui ht0+1; jh t;

d ; wi;t; mi;t; yi;t = E Ui(ht0+1)jhtd because ht

d and h t;

d induce identical continuation games in periods t+ 1 onwards, and all other variables do not a¤ect the continuation game. Therefore, agent i is willing to pay so long as (1 ) _i;t E Si ; ht0+1 h t d; i;t; yi;t E Ui ht0+1 h t d :

i;t depends only on variables that agent i has observed or can perfectly infer from mi;t. Plugging in _i;t, agent i is willing to pay so long as

Bi i;t; yi;tjh t d E Si ; ht0+1 h t d; i;t; yi;t E Si ; ht0+1 h t d; i;t; yi;t E Ui ht0+1 h t d

or Bi i;t; yi;tjhtd E Ui ht0+1 htd . This inequality holds by (3). O¤ the equilibrium path, agent i’s payo¤ is independent of _i;t and so he chooses _i;t = 0. So agent i has no pro…table deviation from _i;t, regardless of his beliefs about the true history.

We conclude that is an RE with the desired properties.

Proof of Proposition 1

7.0.1 Proof of Statement 2

De…nition 1. De…ne the transformation

Gi yij ; di;d~i; ei;e~i =Fi 1 Fi yij ;e~i;d~i ; ei; di :

The distribution over outcomesyi has full support, soFi is strictly increasing and henceFi 1 is a function. F_i 1 is continuously di¤erentiable, because Fi is continuously di¤erentiable. Claim 1. The distribution over Gi yij ; di;d~i; ei;~ei induced by ;d~i;e~i is identical to the distribution overyi induced by ( ; di; ei): Gi(yi)j ;d~i;e~i

d

= yij ; di; ei. Proof of Claim 1. To prove the claim, it su¢ ces to show that for every yi,

F yij ;d~i;~ei =Fi Gi yij ; di;d~i; ei;~ei ; di; ei . This is true by de…nition of Gi.

De…nition 2. Fix a distribution overi-dyad surplus. De…ne

ei( ; di; ) = argmax ei

E[yij ; di; ei] c(ei)

subject to: there exists a mapping Si : [0;1) ! R and a reward scheme Bi : [0;1) ! R satisfying

1. E¤ort IC: ei 2argmaxeifE[Bi(yi)j ; di; ei] c(ei)g

2. Dynamic enforcement: for all yi 2[0;1), Ui( ) Bi(yi) Si(yi) 3. Distribution-matching: Sij ; di; ei

d = .

De…nition 3. For monotonically increasing Si : [0;1) ! R, de…ne e^i ; di;d~i; ei Si implicitly by c0(^ei) = Z y_i( ;di;ei) 0 Ui( ) @fi @ei yij ;d~i;e^i dyi (1) + Z 1 y_i( ;di;ei) Si Gi yij ; di;d~i; ei;e^i @fi @ei yij ;d~i;^ei dyi.

Claim 1. Suppose ; di;d~i; ei are such that di = ~di ande^i( ; di; di; eijSi) =ei. Then^ei is di¤erentiable on a neighborhood about that point.

Proof of Claim 1. Take Si : [0;1)! R to be a monotonically increasing function. The equation (1) may be rewritten as H= 0, where

H Z y_i( ;di;ei) 0 Ui( ) @fi @ei yij ;d~i;^ei dyi + Z ₁ Gi(y_i( ;di;ei)) Si(yi) @G_i 1 @yi @fi @ei G_i 1(yi) ;d~i;^ei dyi c0(^ei):

This expression is continuously di¤erentiable in both d~i and e^i. Therefore, by the Implicit Function Theorem, @^ei

@d~i exists on a neighborhood about ; di;

~

di; ei as long as @H_@e^i 6= 0. To show that this is the case, we will bound H from above by a function H that is di¤erentiable in ^ei and strictly decreasing in e^i on a neighborhood about ; di;d~i; ei , and coincides withH ate^i =ei. For " >0, let

H Z y_i( ;di;ei) 0 Ui( ) @fi @ei (yij ; di;e^i)dyi + Z y_i( ;di;ei)+" y_i( ;di;ei) Si(Gi(yij ; di; di; ei;e^i)) @fi @ei (yij ; di;^ei)dyi + Z 1 y_i( ;di;ei)+" Si(yi) @fi @ei (yij ; di;e^i) c0(^ei).

At^ei =ei,Gi(yi) =yi, soH =H. For ^ei > ei su¢ ciently close, we claim thatH H. Note that Gi(yij ; di; di; ei;e^i) yi if e^i ei, because Fi is FOSD increasing in ei. Therefore,

Si(Gi(yij ; di; di; ei;^ei)) Si(yi), because Si is monotonically increasing. Further, for ^ei su¢ ciently close to ei, @f_@ei_i (yij ; di;^ei) 0 for yi yi ( ; di; ei) +", because @f_@eii is strictly monotonically increasing in yi and equals0 atyi ( ; di; ei) for decision di and e¤ortei. This proves thatH H.

If" = 0, then @H_@e^

i <0by CDFC. @H

@^ei is continuous in", becauseGi is continuous inyi, so @H

@^ei <0 for " su¢ ciently close to 0. SoH is such that H =H atei = ^ei,H H for ^ei > ei

su¢ ciently close, and @H_@e^

i <0. We conclude that @H @e^i <0.

Claim 2. Let be a surplus-maximizing equilibrium. Then for any on-path history ht d 2

Ht

d, and agent i, let i ( jhtd) be the distribution Si ht0+1 htd induced by . Then ei;t =

ei t; di;t; i ( jhtd) .

Proof of Claim 2. Suppose e_i;t > ei t; di;t; i ( jhtd) . It is easy to see that (IC) is a necessary condition for a credible reward scheme to exist. So e_i;t satis…es (2) and (3) by

Lemma 1 and induces a distribution over continuation dyad-surplus equal to _i ( _jht

d). So it must be thate_i;t > eF B

i ( t; di;t)by de…nition ofei. By De…nition 2, there exists some mapping

G(yi;t) such that the distribution Si G(yi;t) htd; ei;t t; di;t; i ( jhtd) equals i ( jhtd). De…ne ~ as a strategy that is identical to except following history ht

d. At history

ht_d, agents j ₆= i play as in . Agent i chooses e¤ort ei t; di;t; i ( jhtd) . Following the realization of output yt, agent i’s output yi;t is treated as output G(yi;t), but otherwise continuation play is identical to . Following ht₀+1, ~ has a credible reward scheme, be- cause does. In period t, there exists a credible reward scheme that induces agent i to choose ei t; di;t; i ( jhtd) by De…nition 2. Agents j 6= i face marginal distributions over continuation payo¤s that are identical to , so they are willing to choose e_j;t as in . At histories ht₀0 for t0 < t, Sj ~; ht 0 0 = Sj ; ht 0 0 for j 6= i, while Si ~; ht 0 0 Si ; ht 0 0 , becauseei t; di;t; i ( jhdt) leads to strictly higher surplus at history htdthan ei;t. So ~ has a credible reward scheme and so is payo¤-equivalent to an equilibrium, which contradicts being surplus-maximizing.

If e_i;t < ei t; di;t; i ( jhdt) , then ei;t < eF Bi ( t; di;t) as well. Then the alternative equilibrium that is identical to except that agenti choosesei t; di;t; i ( jhtd) leads to a strictly higher total surplus, which is a contradiction. Thereforee_i;t =ei t; di;t; i ( jhtd) . Claim 3. ei( t; di;t; i ( jhtd)) is weakly increasing indi;t.

Proof of Claim 3. Fix d_i and di > di. By assumption, for any di di, there exists a

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