Deadline Effects in a Competitive Bargaining Model Simon Board Jeff - - PowerPoint PPT Presentation

Introduction Model Analysis Alternating The End

Deadline Effects in a Competitive Bargaining Model

Simon Board Jeff Zwiebel

UCLA Stanford

March, 2014

Introduction Model Analysis Alternating The End

Motivation

Competitive bargaining

◮ Two agents must agree to a proposed outcome. ◮ Agents can spend resources to influence the negotiations.

Our setting

◮ Agents endowed with limited resources. ◮ Must agree prior to a deadline.

Examples

◮ Political parties negotiating over the debt ceiling. ◮ Groups in a department disagreeing over who to hire. ◮ Sony and Toshiba bargaining over DVD Standard.

Introduction Model Analysis Alternating The End

Main Findings

Dynamic tradeoff

◮ Spend resources to control agenda today? ◮ Or, save capital for future negotiations?

General game

◮ Off-path, competition escalates over time. ◮ On-path, the game generically ends in one period. ◮ Bargaining becomes efficient as the game becomes long.

Example: Agents’ endowments are similar to pie

◮ Competition doubles each period. ◮ Extra ǫ of capital yields extra ǫ/2 of pie. ◮ Natural tie-break yields alternating offers as equilibrium result.

Introduction Model Analysis Alternating The End

Literature

Yildirim (2007)

◮ Agents pay c(xi) to be recognized with probability xi/ j xj. ◮ Infinite periods, exogenous discounting. ◮ Characterize stationary equilibria.

Bargaining games

◮ Rubinstein (1982). ◮ Perry and Reny (1987).

Agenda control in political economy models

◮ Levy and Razin (2014). ◮ Copic and Katz (2012).

Introduction Model Analysis Alternating The End

Model

Introduction Model Analysis Alternating The End

Model

Two agents bargain over pie of size 1

◮ Time finite and indexed backwards, {T, . . . , 1}. ◮ Agents endowed with capital (k1, k2).

Bidding stage

◮ Agents bid in first-price auction. ◮ Payment made to third party or wasted.

Bargaining stage

◮ Winner of auction makes bargaining offer to loser. ◮ If they reject, enter next period with winner poorer. ◮ If reject in period T, agents get (0, 0).

We look for the SPE.

Introduction Model Analysis Alternating The End

Model Details

Continuation utility, net of capital

Ui(kt

1, kt 2, t) = Et

• sτ

i − τ

• r=t

br

i I{i wins in period r}

• Tie-break rules

◮ Indifferent tie. Both i and i have same preferences over

winning/losing, and are indifferent at bid b∗.

◮ Discontinuous tie. Both i and j prefer to win at b ≤ b∗, but

neither wish to win at b > b∗.

◮ Asymmetric tie. Both agent i and j prefer to win at b∗, but i

prefers to win at b∗ + ǫ, while j does not.

Introduction Model Analysis Alternating The End

Analysis of the General Model

Introduction Model Analysis Alternating The End

Bargaining Offers are Efficient

Lemma 1. In any SPE,

U1(kt

1, kt 2, t) + U2(kt 1, kt 2, t) = 1 − bt.

Idea

◮ Agents bargain away future inefficiency.

Implications

◮ Winning agent holds opponent to outside option and extracts

value from ending game earlier.

◮ Future bids act like endogenous discount factor.

Introduction Model Analysis Alternating The End

Equilibrium Bids

◮ If 1 loses, she is held to her outside option

U lose

1

= U1(kt

1, kt 2 − b, t − 1). ◮ If 1 wins, she holds 2 to his outside option,

U win

1

= 1 − U2(kt

1 − b, kt 2, t − 1) − b,

= U1(kt

1 − b, kt 2, t − 1) + bt−1(kt 1 − b, kt 2) − b. ◮ If utility continuous, both agents indifferent at equilibrium bid

1 − U2(kt

1 − b, kt 2, t − 1) − b = U1(kt 1, kt 2 − b, t − 1).

(*)

Introduction Model Analysis Alternating The End

Some Useful Properties

Definitions

◮ Diagonal, D := {(k1, k2) : k1 = k2, k1 ≤ 1/2} ◮ Zero bid region, Z := D ∪ {(k1, k2) : k1 = 0 or k2 = 0}.

Lemma 2. Suppose t ≥ 2.

(a) Ui(kt

i, kt j, t) is increasing in kt i and decreasing in kt j.

(b) Ui(kt

i, kt j, t) is continuous in (kt i, kt j) except at D.

(c) bt(kt

i, kt j) is continuous everywhere.

(d) Bids are zero if and only if (kt

1, kt 2) ∈ Z

(e) If t ≥ 3, (kt

1, kt 2) ∈ Z if and only if (kt−1 1

, kt−1

2

) ∈ Z. (f) If t ≥ 3, equilibrium bids are given by indifference eqn (*).

Introduction Model Analysis Alternating The End

Immediate Termination

Proposition 1. If (kt

1, kt 2) ∈ Z then the game ends in one period.

If (kt

1, kt 2) ∈ Z ◮ Then (kt−1 1

, kt−1

2

) ∈ Z and bt−1 > 0.

◮ There is real cost of bargaining and game ends immediately.

If (kt

1, kt 2) ∈ Z ◮ There is eqm where bt = 0 and game ends immediately. ◮ The is eqm where all offers are rejected and agree in t = 1.

Introduction Model Analysis Alternating The End

Uniqueness

Proposition 2. Payoffs and bidding expenditure are uniquely

determined in equilibrium, independent of the tie-breaking rule.

Idea

◮ Indifference eqn (*) uniquely defines bids. ◮ Agents indifferent, so do not care how ties are broken. ◮ Tie-break rule does affect the distribution of capital off-path.

Introduction Model Analysis Alternating The End

Escalation of Competition

Proposition 3. In any SPE profile, bt−1 ≥ bt. Intuition

◮ In period t, agents fight over waste in next period, bt−1. ◮ Winning also worsens an agent’s future bargaining position.

Proof

◮ If agent 1 wins in t, efficient bargaining means

bt−1 = 1 − U1(kt

1 − bt, kt 2, t − 1) − U2(kt 1 − bt, kt 2, t − 1). ◮ At time t, bt given by indifference eqn (*),

−bt = U1(kt

1, kt 2 − bt, t − 1) + U2(kt 1 − bt, kt 2, t − 1) − 1. ◮ Summing these,

bt−1 − bt ≥ U1(kt

1, kt 2 − bt, t − 1) − U1(kt 1 − bt, kt 2, t − 1) ≥ 0.

Introduction Model Analysis Alternating The End

Efficiency as T → ∞

Proposition 4.

As T → ∞, the initial bid converges to zero at rate bT = O(1/T). Hence, U1(k1, k2, T) + U2(k1, k2, T) → 1.

Idea

◮ Bids escalate over time but sum to less than k1 + k2.

Rent dissipation fails

◮ Posner: The cost of obtaining a monopoly equals the profit of

being a monopolist.

◮ If (k1, k2) ≥ (1, 1), this is true if game is short. ◮ As T → ∞, agents wish to save capital for future bargaining.

Introduction Model Analysis Alternating The End

Even Poor Agents Get Some Pie

Proposition 5. If k1 > 0 then ∃T ∗ such that U1(k1, k2, T) > 0

for T ≥ T ∗.

Idea

◮ If agent 1 is held to zero utility then bt = kt 1/t. ◮ This can be seen by induction, using indifference eqn (*)

bt = kt

1 − bt

t − 1 ⇒ bt = kt

1

t

◮ But agent 2 can’t bid more that this harmonic series forever.

Introduction Model Analysis Alternating The End

Different Regions

Agents similar but poor

◮ Bids very low until final period. ◮ Wealthier agent doesn’t want to give away advantage.

Agents similar but wealthy

◮ In short game, bids equal entire pie. ◮ In long game, resource constraints suppress bids.

One wealthy agent

◮ In short game, wealthy agent takes entire pie. ◮ In longer game, poor agent gets something.

Introduction Model Analysis Alternating The End

Example: Alternating Offer Region

Introduction Model Analysis Alternating The End

Alternating Offer Region

Consider region around (k1, k2) = (1, 1)

◮ Shows tradeoff between agenda control and saving resources. ◮ Solve for utilities in closed form. ◮ Alternating offers is an equilibrium result.

Defining (k2

1, k2 2) ∈ At ◮ A2 defined s.t. if 1 wins in t = 2 then k1 1 ≤ min{k1 2, 1}. ◮ At defined s.t. (kt−1 1

, kt−1

2

) ∈ At−1 under the equilibrium bid.

Introduction Model Analysis Alternating The End

Equilibrium Bids and Utilities

Proposition 6.

Suppose (kt

1, kt 2) ∈ At for t ≥ 2. Then bids are

bt(kt

1, kt 2) = kt 1 + kt 2 − 1

2t − 1 , and agent 1’s continuation utility is U1(kt

1, kt 1, t) = 2t−1 − 1

2t − 1 kt

1 − 2t−1

2t − 1kt

2 + 2t−1

2t − 1.

Implications

◮ Isobid property: transferring ǫ from 1 to 2 does not affect bids. ◮ Transfer property: transferring ǫ from 1 to 2 raises U2 by ǫ. ◮ Doubling property: bids double each period.

Introduction Model Analysis Alternating The End

Why Doubling?

Intuition

◮ Winning costs bt directly and transfers bt to opponent. ◮ Winning enables agent to capture tomorrow’s loss, bt−1.

Proof

◮ If agent 1 wins, the transfer property implies

U win

1

(kt

1, kt 2, t) = U1(kt 1 − bt, kt 2, t − 1) + bt−1 − bt

= U1(kt

1, kt 2 − bt, t − 1) + bt−1 − 2bt. ◮ If agent 1 loses,

U lose

1

(kt

1, kt 2, t) = U1(k1, k2 − bt, t − 1). ◮ Indifference and the isobid property implies

2bt = bt−1.

Introduction Model Analysis Alternating The End

Implications

◮ Winner of t has less capital in t − 1 if offer rejected.

Hence, greater capital tie-break rule yields alternating offers.

◮ If (kT 1 , kT 2 ) = (1, 1) then,

bt = 2T−t 2T − 1 and U1(1, 1, T) = 2T−1 − 1 2T − 1 .

◮ As T → ∞ then,

bT = O(1/2T ) and U1(k1, k2, ∞) = 1 2(k1 − k2) + 1 2.

◮ Formally At is characterized by

kt

1 ≤ min

2tkt

2 − 1

2t − 2 , 2t−2 3 · 2t−2 − 1kt

2 + 1

• .

Introduction Model Analysis Alternating The End

Conclusion

Competitive bargaining model

◮ Agents spend resources to influence negotiations. ◮ Agents must decide prior to a deadline.

Characterizing equilibrium

◮ Competition escalates over time. ◮ Bargaining is efficient as T → ∞. ◮ Bargaining shares depend on initial resources (k1, k2).

What else?

◮ Similar escalation with all-pay auction. ◮ Asymmetric information induces long delays. ◮ Other competitive bargaining models. . .