Continuous Cheap Talk Felix Munoz-Garcia Strategy and Game Theory - - PowerPoint PPT Presentation

Continuous Cheap Talk

Felix Munoz-Garcia Strategy and Game Theory Washington State University

Environment set up

The sate of the world θ is uniformly distributed on [0, 1] Player 1’s action a1, player 2’s action a2, a1, a2 2 [0, 1] Player 2’s payo¤: v2 (a2, θ) = (a2 θ)2 Player 1’s payo¤: v1 (a2, θ) = (a2 (θ + b))2 (where b > 0 is the bias of player 1) Implies player 2’s optimal choice is a2 = θ, player 1’s optimal choice is a2 = θ + b (! …gure)

Payo¤s in the continuous Cheap Talk

No fully truthful equilibrium

Claim: As in …nite game, There can never be a fully truthful equilibrium in this continuous cheap talk game. Proof: If player 2 believes that player 1 reports the true θ, then player 2’s best response is a2 = θ. But if player 2 takes this action, player 1 will report a1 = θ + b to get higher utility.

Babbling perfect Bayesian equilibrium

Claim: There exists a babbling perfect Bayesian equilibrium in which player 1’s message reveals no information and player 2 chooses an action to maximize his expected utility given his prior belief.

Babbling perfect Bayesian equilibrium

Proof: Let player 1’s strategy be a1 = aB

1 2 [0, 1] regardless of θ.

This means player 1’s message is completely uninformative. Player 2 believes that θ is distributed uniformly on [0, 1] max

a2 Ev2 (a2, θ) =

Z 1

0 (θ a2)2 dθ = 1

3 + 2a2 a2

2

which is maximized when a2 = 1

2.

Let player 2’ o¤-equilibrium-path beliefs be Pr

• θ = 1

2

• a1 6= aB

1

= 1, so that his best response to any

• ther message
• a1 6= aB

1

• is a2 = 1

2 as well

From player 1’s payo¤ function, player 1 is indi¤erent between any of his messages and hence best response is a1 = aB

1 .

Two message perfect Bayesian equilibrium

Since there is no truthful equilibrium and there is always a babbling equilibrium, how much information can player 1 credibly transmit to player 2? We begin by constructing a perfect Bayesian equilibrium in which player 1 uses one of two message, a0

1 and a00 1, and player

2 chooses a di¤erent action following each message, a2 (a0

1) < a2 (a00 1 )

Two message perfect Bayesian equilibrium

Claim: In a two-message equilibrium, with the condition a2 (a0

1) < a2 (a00 1 ) , player 1 must use a threshold strategy as

follows: if 0 θ θ he chooses a0

1, whereas if θ θ 1

he chooses a00

1.

Two message perfect Bayesian equilibrium

Proof: player 1’s payo¤s from a0

1 and a00 1 are as follows:

v1

• a2
• a0

1

• , θ
• =
• a2
• a0

1

b θ 2 v1

• a2
• a00

1

• , θ
• =
• a2
• a00

1

b θ 2

Two message perfect Bayesian equilibrium

Extra gain from choosing a00

1 over a0 1 is equal to

∆v1 (θ) =

• a2
• a00

1

b θ 2 +

• a2
• a0

1

b θ 2 ∂∆v1 (θ) ∂θ = 2

• a2
• a00

1

a2

• a0

1

> 0 for a2

• a00

1

> a2

• a0

1

• which implies if type θ prefers a00

1 to a0 1, which means ∆v1 (θ) > 0,

then a00

1 a0 1 for every θ0 > θ, if type θ prefers a0 1 to a00 1, which

means ∆v1 (θ) < 0, then a0

1 a00 1 for every θ0 < θ.

Hence, there must be some threshold type θ = θ, player 1 is indi¤erent between sending two messages.

Two message perfect Bayesian equilibrium

Graph proof: As we can see, when θ = θ, 1

2(θ + b) and 1 2θ is indi¤erent

for player 1, when θ > θ, player 1 prefers 1

2(θ + b) to 1 2θ,

when θ < θ, player 1 prefers 1

2θ to 1 2(θ + b).

Two message perfect Bayesian equilibrium

Now we know the restrictions applying to player 1’s strategy in a two-message equilibrium. What about player 2’s best strategy. Claim: In any two-message perfect Bayesian equilibrium in which player 1 is using a threshold θ strategy as described in last slide, player 2’s equilibrium best response is a2 (a0

1) = θ 2

and a2 (a00

1 ) = 1θ 2 .

Two message perfect Bayesian equilibrium

Proof: In equilibrium, player 2’s posterior belief following a0

1 is that θ

is uniformly distributed on the interval [0, θ] and his posterior belief following a00

1 is that θ is uniformly distributed on the

interval [θ, 1] Hence, player 2 plays a best response if and only if he sets a2 (a1) = ( a2 (a0

1) = E [θ ja0 1 ] = θ 2

a2 (a00

1 ) = E [θ ja00 1 ] = 1θ 2

Two message perfect Bayesian equilibrium

Claim: A two-message perfect Bayesian equilibrium exists if and only if b < 1

4

Two message perfect Bayesian equilibrium

Proof: When θ = θ, player 1 is indi¤erent between a0

1 and a00 1

v1

• a2
• a0

1

• , θ = v1
• a2
• a00

1

• , θ
• θ

2 b θ 2 = 1 θ 2 b θ 2 For fact that θ

2 < θ < 1θ 2 ,

θ b θ 2 =

• θ b 1 θ

2

Two message perfect Bayesian equilibrium

Solve θ = 1

4 b. In order to have θ > 0, b < 1 4.

To complete the speci…cation of o¤-the-equilibrium-path beliefs, let player 2’s beliefs be Pr n θ = θ

2 ja1 /

2 fa0

1, a00 1 g

• = 1, so that he chooses a2 = θ

2 .

Which causes player 1 is indi¤erent between a0

1 and any other

message a1 / 2 fa0

1, a00 1 g . So his threshold strategy is a best

response.

Two message perfect Bayesian equilibrium

Two message perfect Bayesian equilibrium is better than babbling perfect Bayesian equilibrium for both player 1 and player 2. For player 2, in babbling perfect Bayesian equilibrium, she takes the action 1

2 in all states, so that her payo¤ is

• 1

2 θ

2 . In two message perfect Bayesian equilibrium, her payo¤ is 1

2θ θ

2 for θ 2 [0, θ] , 1

2 (θ + 1) θ

2 for θ 2 [θ, 1] . ????prove bigger utility than no informative equilibrium????? For player 1, in babbling perfect Bayesian equilibrium her payo¤ is 1

2 θ b

• 2. In two message perfect Bayesian

equilibrium, her payo¤ is 1

2θ θ b

2 for θ 2 [0, θ] ,

• 1

2 (θ + 1) θ b

2 for θ 2 [θ, 1] .????prove bigger utility than no informative equilibrium?????

More informative equilibrium

What will happen if b becomes even smaller, i.e., b < 1

4?

Similar to the two message perfect Bayesian equilibrium. Consider that case that player 1 makes one of K reports. When θ 2 [0, θ1] , she reports a11; when θ 2 [θ1, θ2] , she reports a12; when θ 2 [θ2, θ3] , she reports a13; . . . ; when θ 2 [θK 1, θK ] , she reports a1K . For convenience, let θ0 = 0, θK = 1.

More informative equilibrium

Similar to two message perfect Bayesian equilibrium, player 2 plays a best response is 1

2 (θK 1 + θK ) if she observes a1K ..

More informative equilibrium

From the …gure ( an example of three message equilibrium), the strategy is also optimal for player 1. θ1, θ2 are the threshold points.

More informative equilibrium

When θ = θ1, 1

2θ1 and 1 2 (θ1 + θ2) are indi¤erent to player 1;

When θ = θ2, 1

2 (θ1 + θ2) and 1 2 (θ2 + 1) are indi¤erent to

player 1; When θ < θ1, 1

2θ1 1 2 (θ1 + θ2) ; when

θ > θ1, 1

2 (θ1 + θ2) 1 2θ1;

When θ < θ2, 1

2 (θ1 + θ2) 1 2 (θ2 + 1) ; when

θ > θ2, 1

2 (θ2 + 1) 1 2 (θ1 + θ2) .

The value of a1K do not matter as long as no two are same. It is just a signal of the state between θK 1 and θK .

More informative equilibrium

So, we have the general condition: When θ = θK ,

1 2 (θK 1 + θk) and 1 2 (θK + θK +1) are indi¤erent.

By symmetry of player 1’s utility function, θK + b = 1 2 1 2 (θK 1 + θk) + 1 2 (θK + θK +1)

• θK +1 θK = θK θK 1 + 4b

That is the interval between states θK +1 and θK is longer by 4b than the interval between θK and θK 1.

More informative equilibrium

Therefore, we have θ1 + (θ1 + 4b) + + (θ1 + 4 (K 1) b) = 1 Kθ1 + 4b (1 + 2 + + (K 1)) | {z }

2bK (K 1)

= 1 Kθ1 + 2bK (K 1) = 1 If b is small enough that 2bK (K 1) < 1, there is a positive value of θ1 that satis…es the equation. Our three message equilibrium example: b =

1 24, θ1 = 1 3 4b = 1 6, θ2 = 2 3 4b = 1 2.

From 2bK (K 1) < 1, we know K < 1

2

• 1 +

q 1 + 2

b

• .

More informative equilibrium

Player 2’s response a2 as a function of θ:

More informative equilibrium

If b were any smaller, more information equilibrium would exist. The greater the di¤erence between player 1’s and player 2’s preference, b, is, the coarser the information transmitted in the equilibrium. The most informative equilibrium in which both player 1 and player 2 get the highest payo¤. (proof by yourself)

Delegation

Is following player 1’s advice really bad for player 2? NOT really! Under delegation, the distance of utility between the outcome and the receiver’s favorite action is b in every state. Under three message equilibrium, the distance of utility between the outcome and the receiver’s favorite action is distance between the axis and

• 1

2θ1 θ

2 , 1

2 (θ1 + θ2) θ

2 and 1

2 (θ2 + 1) θ

2 in the corresponding state.

Delegation

The light blue area is utility loss under delegation. The light green area is utility loss under three message equilibrium. Utility loss under delegation is smaller than utility loss under three message equilibrium as long as b is not too big.

Delegation

Delegation remains better than the most informative equilibrium as long as b <

p 3 6 0.29(Calculus is needed)

Remember in couple slides ago, if b 1

4 = 0.25 the most

informative equilibrium is not informative at all. Thus, delegation is a better choice than soliciting a report for player 2, the information receiver, if b is player 1’s and player 2’s preferences are close enough.

Application: Information and Legislative Organization

In the U.S. House, committees propose bills to legislature. There are two rules, i.e., "open rule" and "close rule" for legislature to chooses to deal with the bill proposals. Under open rule, any amendment can be made by legislature. Under close rule, legislature may either accept or reject the proposed bill, but may not amend it.

Application: Information and Legislative Organization

The following model is similar to the previous one. State, θ, is uniformly distributed from 0 to 1. Committee observes the true state θ. Legislature cannot

• bserve state θ.

Committee recommends a bill a1 to legislature.

Application: Information and Legislative Organization

Under the open rule, legislature may then choose any bill it wish. Under closed rule, it is restricted to either accept a1 or reject it. If legislature reject the bill, the outcome is the status quo a0. Committee’s payo¤ function: (a2 (θ + b))2 . Legislature’s payo¤ function: (a2 θ)2 . a2 is the bill passed by the legislature. From payo¤s function, committee’s favorite bill is always higher than legislature by b. b is a …xed positive number.

Open rule

Under open rule, the model is identical to more informative equilibrium. Committee is player 1 and legislature is player 2. We already know the most informative equilibrium is the best equilibrium.

Closed rule

Under closed rule, legislature retains the power to veto the bill a1 proposed by committee, but not otherwise to amend. In this case, the outcome is the …xed status quo a0. Hence, the outcome is either a1 or a0. The closed rule is more close to delegation. Does the legislature bene…t from retaining this vestige of control?

Closed rule

Closed rule

Claim: In no equilibrium does the committee propose its favorite bill in all states Proof: If committee proposes its favorite bill a1 = θ + b for all θ, the legislature optimally exercise its option whenever θ 2 [a0 b, a0 + b] .or a0 2 [θ b, θ + b] (…gure). Hence a1 = θ + b is not optimal for θ 2 (a0, a0 + b) for committee:

θ + b leads to a0, and (a0 (θ + b))2 < b2. If committee proposes θ + 2b, then legislature accepts. Its payo¤: (θ + 2b (θ + b))2 = b2

Closed rule

Closed rule

Claim: If the status quo a0 2 (b, 1 3b) , the equilibrium that the bills proposed by committee is accepted by legislature is 8 > > > > < > > > > : θ + b if θ a0 b a0 if a0 b < θ a0 a0 + 2b if a0 < θ a0 + 2b a0 + 4b if a0 + 2b < θ a0 + 3b θ + b if a0 + 3b < θ

Closed rule

Graph:

Closed rule

So if any recommendation of less than a0 or greater than a0 + 4b,legislature can infer the state precisely For a0, a0 + 2b and a0 + 4b, committee makes each of these recommendations for an interval of states.

Closed rule

Given distribution of states is uniform, legislature’s belief is 8 > > > > < > > > > : θ = a1 b if a1 a0 uniform distribution of θ 2 [a0 b, a0] if a1 = a0 uniform distribution of θ 2 [a0, a0 + 2b] if a1 = a0 + 2b uniform distribution of θ 2 [a0 + 2b, a0 + 4b] if a1 = a0 + 4b θ = a1 b if a0 + 4b < a1 Assume legislature believes the state is a0 if a1 2 (a0, a0 + 2b) or a1 2 (a0 + 2b, a0 + 4b) .

Note: Following its strategy, committee does not make these strategies.

Therefore, legislature’s strategy is to accept all recommendations except a1 2 (a0, a0 + 2b) and a1 2 (a0 + 2b, a0 + 4b) .

Closed rule

Given legislature’s strategy, committee’s strategy is optimal. If θ a0 b or θ > a0 + 3b, the outcome is committee’s favorite bill.

No other strategy yields a better payo¤.

When θ = a0, then a1 = a0, and payo¤ is b2.

if a1 2 (a0, a0 + 2b) or a1 2 (a0 + 2b, a0 + 4b) , then a2 = a0. if a1 < a0, than a2 = a1 < a0. if a1 = a0 + 2b or a1 = a0 + 4b, than a2 = a0 + 2b or a2 = a0 + 4b From the …gure in next slide, none of these changes increase committee’s payo¤. When θ = a0, a1 = a0 is optimal for committee

Similar analysis for other cases of θ.

Closed rule

Closed rule

Now consider optimality of legislature’s action. When a1 < a0, or a1 > a0 + 4b, legislature knows the true state is a1 b.Then, a1 is preferred to a0. When a1 = a0, legislature has no other option than a2 = a0. When a1 = a0 + 2b, legislature’s belief is θ is uniformly distributed from a0 to a0 + 2b. !…gure

Closed rule

Solid blue line denotes payo¤ as a function of states if a1 is accepted. Dashed blue line denotes payo¤ as a function of states if a1 is rejected. The two lines are mirror images of each other. So legislature is indi¤erent between accepting and rejecting a1, when a1 = a0 + 2b.

Closed rule

When a1 = a0 + 4b, legislature’s belief is θ is uniformly distributed from a0 + 2b to a0 + 3b.

Closed rule

Solid line denotes payo¤ as a function of states if a1 is accepted. Dashed line denotes payo¤ as a function of states if a1 is rejected. Solid line is entirely above the dashed line So legislature prefers accepting to rejecting a1, when a1 = a0 + 4b.

Closed rule

When a1 2 (a0, a0 + 2b) and a1 2 (a0 + 2b, a0 + 4b) , legislature believes that the state is a0, it is optimal to reject a1.

Comparison

Open rule and close rule Open rule Close rule We know the more close is outcome to legislature’s favorite bills, θ, the better for its bene…t. Equilibrium under close rule is better than equilibrium under

• pen rule.

Comparison

For committee, close rule is also better than open rule. Because close rule gives the committee an incentive to divulge the information in limiting legislature’s ability to take advantage of it.

Comparison

Close rule and delegation From committee’s point of view, committee wants outcome more close to its favorite bills, θ + b. So delegation is better than close rule for committee.

Comparison

The shadowed area shows the deviation of the bill proposed by committee from the legislature’s favorite bill under close rule. The area between line of b and horizontal axis is the deviation under delegation from the legislature’s favorite bill. The two area are equal. So legislature is indi¤erent between close rule and delegation. XXXXdi¤erent from corresponding part in last paragraph page 356 Osborne.XXXXX

Comparison

For legislature, close rule is better than open rule, Delegation generates an even better outcome than under close rule.(contradicted by last slide) Legislature’s best option is to relinquish all control over

• utcome and to cede the choice of legislation to the

committee.