dynamic games, ii: bargaining › ~tjohns20 › gametheory_fa19 › slides5.pdf · econ 400 (nd)...

$Page 1: Dynamic Games, II: Bargaining › ~tjohns20 › gametheory_FA19 › slides5.pdf · Econ 400 (ND) Dynamic Games, II: Bargaining 2 / 13. The \Dictator" Game This is a very \stark" example$

Dynamic Games, II: Bargaining

Econ 400

University of Notre Dame

Econ 400 (ND) Dynamic Games, II: Bargaining 1 / 13

Bargaining

In bargaining games, we are generally trying to model agents “haggling”back-and-forth over how to split the value or “surplus” from trade. Theytypically have the following ingredients:

There’s a “pie”/value/surplus of fixed value. We usually normalize itto 1, so the players are trying to get the largest share they can. Theplayers can choose any split, (s, 1− s).

There’s an extensive form that describes who makes offers andcounter-offers when, and under what circumstances bargaining“breaks down” and the players walk away from the tradingopportunity.

If trade breaks down, the players get their outside option payoffs, ri ,which represent what the player can get on his own. We usuallyassume that r1 + r2 < 1 in two-players bargaining games, so thattrade would be efficient if it can be arranged.

The “Dictator” Game

This is a very “stark” example of a bargaining game:

Player one proposes a split of the pie, (t, 1− t), with t going toplayer 1.

Player two accepts or rejects. If player 2 accepts, they split the pie.Otherwise, player 2 gets r2 and player 1 gets r1, where r1 + r2 < 1.

What’s the subgame perfect Nash equilibrium, especially whenr1 = r2 = 0? Is this a good prediction?

The “Dictator” Game

This is a very “stark” example of a bargaining game:

Player one proposes a split of the pie, (t, 1− t), with t going toplayer 1.

Player two accepts or rejects. If player 2 accepts, they split the pie.Otherwise, player 2 gets r2 and player 1 gets r1, where r1 + r2 < 1.

What’s the subgame perfect Nash equilibrium, especially whenr1 = r2 = 0? Is this a good prediction?

Discounting

Suppose an agent gets a payoff u for each date t = 0, 1, 2, 3, ...,T . Thenthe discounted payoff when an agent has discount factor δ is given by thesum

S = u + δu + δ2u + ...+ δTu

where δ < 1.

Alternating Offer Bargaining

But suppose that rather than accept t∗ = r2, the second player makes acounteroffer. To make it interesting, we imagine that there are costs todelay, so that the players’ payoffs after a counteroffer are discounted by δ,giving us an extensive form:

In particular, if r1 = r2 = 0, what is the SPNE?

Or...

If we set r1 = r2 = 0, a pattern begins to emerge: The game ends in thefirst round, with the first player offering

t11 = 1

t21 = 1− δ

t31 = 1− δ(1− δ) = 1− δ + δ2

t41 = 1− δ(1− δ(1− δ)) = 1− δ + δ2 − δ3

...

tT1 = 1− δ + δ2 − δ3 + δ4 − ...± δT−1 = ΣTt=0(−δ)t

But these are somewhat confusing because of the ... terms.

If we set r1 = r2 = 0, a pattern begins to emerge: The game ends in thefirst round, with the first player offering

t11 = 1

t21 = 1− δ

t31 = 1− δ(1− δ) = 1− δ + δ2

t41 = 1− δ(1− δ(1− δ)) = 1− δ + δ2 − δ3

...

tT1 = 1− δ + δ2 − δ3 + δ4 − ...± δT−1 = ΣTt=0(−δ)t

But these are somewhat confusing because of the ... terms.

Geometric Sums

But how do we compute a sum like S = y + xy + x2y + ...+ xT y? Firstmultiple S by x :

xS = xy + x2y + x3y + ...+ xT+1y

Then

S = y +xy + x2y + x3y + ...+ xT y

−xS = −xy − x2y − x3y − ...− xT y - xT+1y

S(1− x) = y - xT+1y

Or

S(T ) =1− xT+1

1− xy

Note that

S(∞) =1

1− xy

Geometric Sums

But how do we compute a sum like S = y + xy + x2y + ...+ xT y? Firstmultiple S by x :

xS = xy + x2y + x3y + ...+ xT+1y

Then

S = y +xy + x2y + x3y + ...+ xT y

−xS = −xy − x2y − x3y − ...− xT y - xT+1y

S(1− x) = y - xT+1y

Or

S(T ) =1− xT+1

1− xy

Note that

S(∞) =1

1− xy

Then using the geometric summation formula on

tT1 = 1− δ + δ2 − δ3 + δ4 − ...± δT−1 = ΣTt=0(−δ)t

yields

tT1 = ΣT−1t=0 (−δ)t =

1− (−δ)T

1 + δ

If we take the limit as we allow the bargaining to go on indefinitely,T →∞, and

t∞1 =1

1 + δ

and the players’ payoffs are

1

1 + δ,

δ

1 + δ

Then using the geometric summation formula on

tT1 = 1− δ + δ2 − δ3 + δ4 − ...± δT−1 = ΣTt=0(−δ)t

yields

tT1 = ΣT−1t=0 (−δ)t =

1− (−δ)T

1 + δ

If we take the limit as we allow the bargaining to go on indefinitely,T →∞, and

t∞1 =1

1 + δ

and the players’ payoffs are

1

1 + δ,

δ

1 + δ

So as δ → 1, we get

limδ→1

t∞ = limδ→1

1

1 + δ=

1

2

so the players split the value equally.

Likewise, if δ = e−R∆, so thatinterest compounds continuously at rate R over a short period of time oflength, ∆

lim∆↓0

t∞ = lim∆↓0

1

1 + e−R∆=

1

2

So if the players become very patient, or the time between offers goes tozero, the players split the pie equally: Bargaining power here comes from“agenda” control and the ability to waste resources credibly. If these areremoved by allowing many offers and counter-offers, or the time betweenoffers goes to zero, then the players split the pie equally in the SPNE.

limδ→1

t∞ = limδ→1

1

1 + δ=

1

2

so the players split the value equally. Likewise, if δ = e−R∆, so thatinterest compounds continuously at rate R over a short period of time oflength, ∆

lim∆↓0

t∞ = lim∆↓0

1

1 + e−R∆=

1

2

limδ→1

t∞ = limδ→1

1

1 + δ=

1

2

so the players split the value equally. Likewise, if δ = e−R∆, so thatinterest compounds continuously at rate R over a short period of time oflength, ∆

lim∆↓0

t∞ = lim∆↓0

1

1 + e−R∆=

1

2

What’s missing from this model of bargaining that is probably importantin real applications?

The Hold-Up Problem

In many situations, opportunities to bargain and the inability to commitlead to inefficient outcomes, even when property rights are clearly defined.

The student decides on an amount of education to get, e ≥ 0, whichcosts c

2e2

The firm makes the student a wage offer, w ≥ 0

The student can accept the job, receiving a payoff of w − c2e

2, orreject, and open his own business that makes profits be. If thestudent accepts the job, the firm makes profits ae, where a > b, andotherwise the firm makes nothing.

Sketch an extensive form for this game and solve for the subgame perfectNash equilibrium. What is the efficient level of education? Is the studentover- or under-educated in the SPNE relative to the efficient outcome? Ifthe firm could tie its hands and commit to making a wage offerw = a ∗ (a/c), would the problem be resolved?

The Hold-Up Problem

2e2

The Hold-Up Problem

2e2

Partnership in Firms

Instead of paying a wage w , imagine that the firm offers a share 0 < s < 1of the profits to the student contingent on getting a degree, like becominga partner in a law firm or a medical practice. How high does the sharehave to be to get the student to take the job, and what level of educationdoes he choose?

dynamic games, ii: bargaining › ~tjohns20 › gametheory_fa19 › slides5.pdf · econ 400 (nd)...

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