Maximization without full revelation

29

Corollary 1.3.

Let Ω = {ω1, . . . , ω_L} and sort A such that A = {a^∗_R(ω₁), . . . , a^∗_R(ω_L)}. We can ignore all actions, which are never the best reply for the Receiver in a single state.

There is a fully revealing equilibrium withˆa=a^∗_R(ω₁)if:

1)u_Shas increasing differences

2)u_S(a^∗_R(ω₂), ω₂)−c > u_S(a^∗_R(ω₁), ω₂)

3)∀ωi∈ {ω2, . . . , ω_L}:u_S(a^∗_R(ω_i), ω_i)≥u_S(a^∗_R(ω_i−1), ω_i)

The fully revealing equilibrium is such that the Sender sends cheap-talk inω1and verifiable mes-sages in all other states.

The threat point here isa^∗_R(ω₁). Condition 2) ensures that the Sender prefers the verifiable message in the state after, which isω₂. Increasing Differences mean that the gains from a higher action increase, if the state gets higher. With Condition 3) combined, we get that the Sender also prefers the verifiable message in all states higher thanω₂. We can get a similar result witha^∗_R(ω_L), where we have to replaceω₂in Condition 2) by ω_L−1 and adjust Condition 3) as well. An application can be found in Section 1.3.1.1.

Similar changes for decreasing differences can be made easily:

Proposition 1.2(Full revelation under decreasing differences).

LetΩandAbe as in Proposition 1.1. There is a fully revealing equilibrium withˆa=a^∗_R(ω_j)if:

1)uS has decreasing differences

2.1)u_S(a^∗_R(ω_L), ω_L)−c > u_S(a^∗_R(ω_j), ω_L) 2.2)u_S(a^∗_R(ω₁), ω₁)−c > u_S(a^∗_R(ω_j), ω₁)

3.1)∀ωi > ω_j :u_S(a^∗_R(ω_i), ω_i)≥u_S(a^∗_R(ω_i+1), ω_i) 3.2)∀ωi < ω_j :u_S(a^∗_R(ω_i), ω_i)≥u_S(a^∗_R(ω_i−1), ω_i)

The fully revealing equilibrium is such that the Sender sends cheap-talk inω_j and verifiable mes-sages in all other states.

Changing the conditions as in Corollary 1.2 and Corollary 1.3 is possible.

30 Chapter 1. Communication Games with Optional Verification cheap-talk messages can have the same effect, but cheap-talk messages can also partial reveal information to the Receiver, such that it splits the state space into disjoint subsets. In that case the Receiver might just know whether he is in the first or second state, or in the third or fourth state.

We give conditions for all the different possibilities of partial revelation and also for combinations of those. Furthermore, we again use utility functions with increasing or decreasing differences to simplify these conditions.

1.2.2.1 Three state examples

Assume|Ω|= 3and assume that the Receiver prefers different actions in different states.

In case of partial revelation the Receiver can maximize his utility by three different possibilities:

1. Use the same action in every state.

2. Reply with one action to one cheap-talk message and with another to the remaining mes-sages.

3. Use the same action as a reply to any cheap-talk message, enforcing the Sender to send the verifiable message in exactly one state, i.e. revelation of one state.

First possibility

maxˆa

X3 i=1

P[ω_i]u_R(ˆa, ω_i)

s.t.∀ωi :uS(ˆa, ωi)> uS(a^∗_R(ωi), ωi)−c Second possibility (Revelation inω₁(wlog) by cheap-talk)

maxˆa

P[ω₁]u_R(a^∗_R(ω₁), ω₁) + X3 i=2

P[ω_i]u_R(ˆa, ω_i) s.t.u_S(a^∗_R(ω₁), ω₁)> u_S(ˆa, ω₁)

u_S(ˆa, ω_i)> u_S(a^∗_R(ω₁), ω_i)∀ωi, i∈ {2,3}

Third possibility (Revelation inω₁(wlog) by a verifiable message) maxˆa

P[ω₁]u_R(a^∗_R(ω₁), ω₁) + X3

i=2

P[ω_i]u_R(ˆa, ω_i) s.t.uS(a^∗_R(ω1), ω1)−c > uS(ˆa, ω1)

uS(ˆa, ωi)> uS(a^∗_R(ωi), ωi)−c∀ωi, i∈ {2,3}

These are the different maximization problems the Receiver has to solve to find the best strategy.

In the second case we do not need to state a condition that the Sender uses the verifiable message, because that condition (similar to the last condition of the third case) is weaker than the last condition of the second case.

With different example we will show that either of the strategies can be the best choice. For that

31 we have to keep in mind that the Receiver cannot commit to any strategies, but plays his best possibility given his beliefs. Especially in the case where he knows the true state, the Receiver will always play the action that yields the highest utility for him.

Example 1.3. Assumec >2,|Ω|=|A|= 3,

uR(a3, ω1) = 4 uR(a1, ω1) = 2 uR(a2, ω1) = 1 u_R(a₁, ω₂) = 4 u_R(a₂, ω₂) = 2 u_R(a₃, ω₂) = 1 u_R(a₂, ω₃) = 4 u_R(a₁, ω₃) = 2 u_R(a₃, ω₃) = 1

andu_S(·, ω1) =u_R(·, ω1), butu_S(·, ω2) =u_R(·, ω3)andu_S(·, ω3) = u_R(·, ω2). So Sender and Receiver have the same preferences inω1, but switched betweenω2andω3.

No full revelation

Clearly there is no full revelation just by cheap-talk messages. The proof why there is no full revelation just by verifiable messages and also not by both message types used, follows the same idea: Assume thatˆa=π₁a₁+π₂a₂+ (1−π₁−π₂)a₃, therefore

uS(a3, ω1)−c > uS(ˆa, ω1) u_S(a₁, ω₂)−c > u_S(ˆa, ω₂) u_S(a₂, ω₃)−c > u_S(ˆa, ω₃) have to hold. Rewriting this yields to

4−c > π₁·1 +π₂·2 + (1−π₁−π₂)·4 2−c > π₁·2 +π₂·4 + (1−π₁−π₂)·1 2−c > π₁·4 +π₂·2 + (1−π₁−π₂)·1 and finally to

c <3π₁+ 2π₂ 1−c > π₁+ 3π₂ 1−c >3π₁+π₂

This is impossible forc≥1. For the full revelation by both message types,ˆacan also be equal to a₁,a₂ora₃, but all these possibilities still contradict at least one condition.

Maximization 1.) No revelation:

The Receiver solvesmax¹₃ ·(2π₁+ 1π₁+ 4(1−π₁−π₂)) +¹₃ ·(4π₁+ 2π₁+ 1(1−π₁−π₂)) +

1

3 ·(2π₁ + 4π₁+ 1(1−π₁−π₂)), which yields to ˆa= a₁ and expected utility for the Receiver

32 Chapter 1. Communication Games with Optional Verification

given byE[u_R] = ¹₃(2 + 4 + 2) = ⁸₃. The conditions for the Sender not to deviate are:

u_S(ˆa, ω₁)> u_S(a₃, ω₁) ⇔ 2>4−c u_S(ˆa, ω₂)> u_S(a₁, ω₁) ⇔ 2>4−c u_S(ˆa, ω₃)> u_S(a₂, ω₁) ⇔ 4>2−c.

All these conditions hold forc >2.

2.) Partial revelation ofω₁by cheap-talk:

The Receiver has to answer witha₃ to one cheap-talk message and with ˆa to the others. The maximization problem yields thatˆa=π₁a₁+ (1−π₁)a₂, withπ₁∈[0,1]. The conditions for the Sender’s utility are

uS(a3, ω1)> uS(ˆa, ω1) ⇔ 4> uS(ˆa, ω1) u_S(ˆa, ω₂)> u_S(a₃, ω₁) ⇔ u_S(ˆa, ω₂)>1 u_S(ˆa, ω₃)> u_S(a₃, ω₁) ⇔ u_S(ˆa, ω₃)>1,

which are clearly satisfied. Here the Receiver’s expected utility isE[uR] = ¹₃(4 + 6) = ¹⁰₃. 3.) Partial revelation ofω₁ by a verifiable message:

For example we can takeˆa=a₂. The conditions for the Sender’s utility are u_S(a₃, ω₁)−c > u_S(a₂, ω₁) ⇔ 4−c >1 u_S(a₂, ω₂)> u_S(a₁, ω₁)−c ⇔ 4>2−c u_S(a₂, ω₃)> u_S(a₂, ω₁)−c ⇔ 2>2−c,

which all hold forc <3. The Receiver’s expected utility isE[u_R] = ¹₃(4 + 6) = ¹⁰₃ .

In this example it is possible to get partial revelation inω₁ either by cheap-talk or by verifiable message ifc∈(2,3). Forc >3partial revelation is just possible by cheap-talk.

Example 1.4. Assumec >2,|Ω|=|A|= 3,

uR(a3, ω1) = 4 uR(a1, ω1) = 2 uR(a2, ω1) = 1 u_R(a₁, ω₂) = 4 u_R(a₂, ω₂) = 2 u_R(a₃, ω₂) = 1 u_R(a₂, ω₃) = 4 u_R(a₁, ω₃) = 2 u_R(a₃, ω₃) = 1 and

u_S(a₃, ω₁) = 4 u_R(a₁, ω₁) = 2 u_R(a₂, ω₁) = 1 u_S(a₃, ω₂) = 4 u_R(a₂, ω₂) = 2 u_R(a₁, ω₂) = 1 u_S(a₃, ω₃) = 4 u_R(a₁, ω₃) = 2 u_R(a₂, ω₃) = 1.

No full revelation

Obviously there is no fully revealing equilibrium just by cheap-talk message, because Sender and Receiver prefer different actions in two states. To see that full revelation just by verifiable messages is impossible, we take a closer look atω₂: To have an incentive to send the verifiable information the Sender has to preferu_s(a₁, ω₂)−coveru_S(ˆa, ω₂). Since the left part equals 1−cand the right part something larger than1, this is impossible. The same arguments contradict the full revelation by both message types for mixed strategies.

33

Forˆa=a₁, the Sender does not use the verifiable message inω₃and forˆa=a₂ she uses cheap-talk inω₂. So there is no full revelation possible.

Maximization 1.) No revelation:

The maximization here is the same as in the previous example. It is possible forc > 2and the expected utility isE[uR] = ⁸₃.

2.) Partial revelation by cheap-talk

Getting partial revelation in ω₁ is impossible, because the Sender will use the same cheap-talk message in both other states. To get partial revelation inω₂,u_S(a₁, ω₂)> u_S(ˆa, ω₂)has to hold, which is impossible for anyˆa 6= a₁. Same arguments work witha₂ inω₃. This means in this example it is not possible to achieve partial revelation by different answers to cheap-talk.

3.) Partial revelation ofω₁ by a verifiable message:

For example we can takeˆa=a₂. The conditions for the Sender’s utility are u_R(a₃, ω₁)−c > u_S(a₂, ω₁) ⇔ 4−c >1 u_R(a₂, ω₂)> u_S(a₁, ω₁)−c ⇔ 2>1−c u_R(a₂, ω₃)> u_S(a₂, ω₁)−c ⇔ 1>1−c,

which again all hold forc <3. The Receiver’s expected utility isE[u_R] = ¹₃(4 + 6) = ¹⁰₃. In this example partial revelation is only possible by verifiable information and just if

c∈(2,3)holds. Forc >3partial revelation is impossible and the Receiver maximizes his utility as he would do without communication.

1.2.2.2 General results

Again we would like to underline that the Receiver cannot commit to any actions, but maximizes his utility. Then it should be obvious that the Receiver always prefers partial revelation over no revelation at all. If there is a partial revelation of one state, the Receiver will maximize his expected utility in the remaining states. It might happen that several actions (pure or mixed) solve this maximization problem.

Definition 1.4. ForΩ^′⊆Ωwe defineA(Ω˙ ^′) = arg max

a

X

ω∈Ω^′

µ[ω]u_R(a, ω).

A(Ω˙ ^′)is the set of actions which maximize the Receiver’s utility on a given state spaceΩ^′ accord-ing to the Receiver’s belief systemµ.

In a general model with more than three states, there can be different types of partial revelation:

Partial revelation can be either achieved by verifiable messages, which then fully reveal a subset of states or by cheap-talk messages. Partial revelation by cheap-talk creates a partition of state subsets, each element of the partition can contain a single state or several states. Elements with just a single state have the same effect as verifiable messages: The Receiver knows whether specific state is the true state. For simplicity we split partial revelation by cheap-talk up into two cases.

34 Chapter 1. Communication Games with Optional Verification 1 Partial revelation by verifiable messages⇒Full Revelation of a subset of states

2 A Partial revelation by cheap-talk⇒Full Revelation of a subset of states

B Partial revelation by cheap-talk⇒Dividing the state space into disjoint subsets.

The case 2A contains just the special cases, in which the partition consists of some subsets with just one element and a subset containing the remaining states. Note that also in case 2 there can be a full revelation of subsets of states.

In a world with four states{ω1, . . . ω₄}partial revelation by type 2B for example means that the Receiver just knows whether the true state is in{ω1, ω₂}or in{ω3, ω₄}. Of course there can also be a combination of type 1 with 2A or with 2B.

Even with just 4 states most often it is impossible to see, which partial revelation is possible without calculating all possibilities. We state conditions for each of the different types and their combinations. With these conditions it is easy to write an algorithm and let a computer check all the possibilities.

Partial Revelation by one message type

Proposition 1.3(Partial Revelation just by Verifiable Messages).

There is a partial revealing equilibrium in which the Sender uses verifiable messages only if the true stateωˆ in an element ofΩ^vI (ΩwithΩ^vI such that:

∃ˆa∈A(Ω˙ \Ω^vI) : 1)∀ω∈Ω^vI :u_S(a^∗_R(ω), ω)−c > u_S(ˆa, ω) 2)∀ω∈Ω\Ω^vI :u_S(a^∗_R(ω), ω)−c < u_S(ˆa, ω).

With this proposition we can define the family of subsets in which partial revelation by verifiable information is possible.

Definition 1.5.

Ω^vI(Ω) =n

Ω^vI (Ω∃ˆa∈A(Ω˙ \Ω^vI)such that

∀ω∈Ω^vI :u_S(a^∗_R(ω), ω)−c > u_S(ˆa, ω)and

∀ω∈Ω\Ω^vI :u_S(a^∗_R(ω), ω)−c < u_S(ˆa, ω) o

This implies that partial revelation by verifiable information is impossible ifΩ^vI(Ω) = {∅}. We can also define the set of all tuples of actions and subsets of states (ˆa,Ω^vI), where the action maximizes the Receiver’s utility onΩ\Ω^vI, but for the states inΩ^vI this action works as a threat to enforce the Sender to use the verifiable message.

Definition 1.6.

Ω^vIA(Ω) =n ˆ

a,Ω^vI 1) ˆa∈A(Ω˙ \Ω^vI)

2)∀ω∈Ω^vI :u_S(a^∗_R(ω), ω)−c > u_S(ˆa, ω)

3)∀ω∈Ω\Ω^vI :u_S(a^∗_R(ω), ω)−c < u_S(ˆa, ω) o

35 This definition will help to combine different types of partial revelation. We can make similar statements for partial revelation of type 2A:

Proposition 1.4(Partial Revelation just by Cheap-Talk).

There is a partial revealing equilibrium in which the Sender uses cheap-talk messages to reveal the true state only if the true stateωˆ is an element ofΩ^ct(ΩwithΩ^ctsuch that:

∃ˆa∈A(Ω˙ \Ω^ct) : 1)∀ω∈Ω^ct:u_S(a^∗_R(ω), ω) > u_S(ˆa, ω) 2)∀ω∈Ω\Ω^ct:u_S(a^∗_R(ω), ω)< u_S(ˆa, ω).

Definition 1.7.

Ω^ct(Ω) =n

Ω^ct(Ω∃ˆa∈A(Ω˙ \Ω^ct)such that

∀ω∈Ω^ct:u_S(a^∗_R(ω), ω)> u_S(ˆa, ω)and

∀ω∈Ω\Ω^ct:u_S(a^∗_R(ω), ω)< u_S(ˆa, ω) o Definition 1.8.

Ω^ctA(Ω) =n ˆ

a,Ω^ct 1) ˆa∈A(Ω˙ \Ω^ct)

2)∀ω ∈Ω^ct:uS(a^∗_R(ω), ω)> uS(ˆa, ω)

3)∀ω ∈Ω\Ω^ct:u_S(a^∗_R(ω), ω)< u_S(ˆa, ω) o For partial revelation of type 2B the conditions look a little bit different.

Proposition 1.5(Partial Revelation just by Cheap-Talk).

There is a partial revealing equilibrium, where the state spaceΩis split up into disjoint subsets if there exists a series of sets(Ω^div_j )_j=1,...,Jsuch that

1. S

j= 1JΩ^div_j = Ωand∀k6=l: Ω^div_k ∩Ω^div_l =∅.

2. ∀Ω^div_j ∃ˆa_j ∈A(Ω˙ ^div_j )such thatu_S(ˆa_k, ω)> u_S(ˆa_l, ω)∀ω∈Ω^div_k withk6=l.

The first condition says that the subsets have to be disjoint and add up to the complete state space.

The second condition ensures that the Sender has no incentive to lie if the Receiver chooses the actions that maximize his expected utility for each subset. As before, we can write this as a set, this time consisting of series of tuples of actions and subsets of the state space:

Definition 1.9.

Ω^divA (Ω) =n ˆ

a_j,Ω^div_j

j

1)∀Ω^div_k :a_k∈A(Ω˙ ^div_k ) 2) [

jΩ^div_j = Ωand∀k6=l: Ω^div_k ∩Ω^div_l =∅ 3)∀ω∈Ω^div_k :∀ˆa_k6= ˆa_l:uS(ˆa_k, ω)> uS(ˆa_l, ω) o

This set contains all the different possibilities of series of tuples that split the state space into subsets.

36 Chapter 1. Communication Games with Optional Verification

Partial revelation by a combination of verifiable messages and cheap-talk

For the combination of two types of partial revelation it is not sufficient to combineΩ^vI andΩ^ct, because we need to use the same actionˆafor the states, that are not revealed.

Theorem 1.4(Partial Revelation by type 1 and 2A).

All combinations of revelation by verifiable message and cheap-talk (type 2A) are given by:

Ω^vI+ct(Ω) =n

Ω^vI,Ω^ct 1) Ω^vI∩Ω^ct=∅

2)∃ˆa∈A˙ Ω\(Ω^vI∪Ω^ct)

such that (ˆa,Ω^vI)∈Ω^vIA(Ω\Ω^ct)and

(ˆa,Ω^ct)∈Ω^ctA(Ω\Ω^vI) o

This means that all states inΩ^vI are revealed by verifiable messages and those inΩ^ct by cheap-talk. Therefore it is necessary that ˆa maximizes the Receiver’s utility for the remaining states Ω\(Ω^vI∪Ω^ct). By the definition ofΩ^vIandΩ^ctit is ensured thatΩ^vI∪Ω^ct(Ωholds, because otherwise there would be full revelation.

Similar to the combination of type 1 and type 2A, it is also possible to combine type 1 and type 2B.

This means that there are some states revealed by verifiable information (type 1) and the remaining states are divided into subsets of the state space (type 2B).

Theorem 1.5(Partial Revelation by type 1 and 2B).

All combinations of revelation by verifiable message and cheap-talk (type 2B) are given by:

Ω^vI+div(Ω) = (

ˆ

a_j,Ω^div_j

j,Ω^vI 1) [

jΩ^div_j

∪Ω^vI = Ω 2)∀Ω^div_k : Ω^div_k ∩Ω^vI =∅ 3) (ˆa_j,Ω^div_j )_j ∈Ω^div_A (Ω\Ω^vI) 4)∀ˆa_k : (ˆa_k,Ω^vI)∈Ω^vI_A

Ω\[

l6=kΩ^div_l ) Condition 1) and 2) ensure that the subsets of states are disjoint, but united are equal to the entire state space. Condition 3) makes sure that the states are split up, if there is no revelation by a veri-fiable message. The Receiver plays different actions on different subsets of states, with Condition 4) the Sender will send the verifiable message in all states inΩ^vI and will not deviate to an action ˆ

a_kfrom the series(ˆa_j).

1.2.2.3 Increasing and Decreasing Differences

For increasing (or decreasing) differences, we can state the existence of partial revealing equilibria with verifiable messages in a way similar to Proposition 1.1. The most important change is that the answer to cheap-talk is no longera^∗_R, butˆasuch that this action maximizes the Receiver’s utility on the non-revealed states.

37

Proposition 1.6(Partial Revelation by verifiable messages under increasing differences).

Let Ω = {ω1, . . . , ω_L} and sort A such that A = {a^∗_R(ω₁), . . . , a^∗_R(ω_L)}. We can ignore all actions, which are never the best reply for the Receiver in a single state.

There is a partial revealing equilibrium that reveals the states just in[ω, ω]by verifiable messages if

∃ˆa∈A(Ω˙ \[ω, ω])witha^∗_R(ω)>ˆasuch that

1)uS has increasing differences onΩ^′ = [ω, ω]andA^′ = [a^∗_R(ω), a^∗_R(ω)]

2)u_S(a^∗_R(ω), ω)−c > u_S(ˆa, ω)

3)∀ω_i ∈[ω, ω] :u_S(a^∗_R(ω_i), ω_i)≥u_S(a^∗_R(ω_i−1), ω_i) 4)∀ωj ∈Ω\[ω, ω] :u_S(ˆa, ω_j)> u_S(a^∗_R(ω_j), ω_j)−c

Proposition 1.7(Partial Revelation by verifiable messages under increasing differences).

Let Ω = {ω1, . . . , ω_L} and sort A such that A = {a^∗_R(ω₁), . . . , a^∗_R(ω_L)}. We can ignore all actions, which are never the best reply for the Receiver in a single state.

There is a partial revealing equilibrium that reveals the states just in[ω, ω]by verifiable messages if

∃ˆa∈A(Ω˙ \[ω, ω])witha^∗_R(ω)<ˆasuch that

1)u_S has increasing differences onΩ^′ = [ω, ω]andA^′ = [a^∗_R(ω), a^∗_R(ω)]

2)u_S(a^∗_R(ω), ω)−c > u_S(ˆa, ω)

3)∀ωi ∈[ω, ω] :u_S(a^∗_R(ω_i), ω_i)≥u_S(a^∗_R(ω_i+1), ω_i) 4)∀ωj ∈Ω\[ω, ω] :u_S(ˆa, ω_j)> u_S(a^∗_R(ω_j), ω_j)−c

We can do a similar change to Condition 3) as before and also get the same results for decreasing differences by the same changes as done between Propositions 1.1 and 1.2.

In addition it is possible to rewrite these conditions that they hold for more than just a single interval[ω, ω], but for a disjoint series of intervals [ω_k, ω_k]

k.

Im Dokument Essays on Communication and Information Transmission (Seite 45-53)