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 ( x i ) to be recognized with probability x i / � j x j . ◮ 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 ( k 1 , k 2 ) . 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 � τ � � U i ( k t 1 , k t 2 , t ) = E t s τ b r i − i I { i wins in period r } r = t 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, U 1 ( k t 1 , k t 2 , t ) + U 2 ( k t 1 , k t 2 , t ) = 1 − b t . 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 = U 1 ( k t 1 , k t 2 − b, t − 1) . 1 ◮ If 1 wins, she holds 2 to his outside option, U win = 1 − U 2 ( k t 1 − b, k t 2 , t − 1) − b, 1 = U 1 ( k t 1 − b, k t 2 , t − 1) + b t − 1 ( k t 1 − b, k t 2 ) − b. ◮ If utility continuous, both agents indifferent at equilibrium bid 1 − U 2 ( k t 1 − b, k t 2 , t − 1) − b = U 1 ( k t 1 , k t 2 − b, t − 1) . (*)
Introduction Model Analysis Alternating The End Some Useful Properties Definitions ◮ Diagonal, D := { ( k 1 , k 2 ) : k 1 = k 2 , k 1 ≤ 1 / 2 } ◮ Zero bid region, Z := D ∪ { ( k 1 , k 2 ) : k 1 = 0 or k 2 = 0 } . Lemma 2. Suppose t ≥ 2 . (a) U i ( k t i , k t j , t ) is increasing in k t i and decreasing in k t j . (b) U i ( k t i , k t j , t ) is continuous in ( k t i , k t j ) except at D . (c) b t ( k t i , k t j ) is continuous everywhere. (d) Bids are zero if and only if ( k t 1 , k t 2 ) ∈ Z 2 ) ∈ Z if and only if ( k t − 1 , k t − 1 (e) If t ≥ 3 , ( k t 1 , k t ) ∈ Z . 1 2 (f) If t ≥ 3 , equilibrium bids are given by indifference eqn (*).
Introduction Model Analysis Alternating The End Immediate Termination Proposition 1. If ( k t 1 , k t 2 ) �∈ Z then the game ends in one period. If ( k t 1 , k t 2 ) �∈ Z ) �∈ Z and b t − 1 > 0 . ◮ Then ( k t − 1 , k t − 1 1 2 ◮ There is real cost of bargaining and game ends immediately. If ( k t 1 , k t 2 ) ∈ Z ◮ There is eqm where b t = 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, b t − 1 ≥ b t . Intuition ◮ In period t , agents fight over waste in next period, b t − 1 . ◮ Winning also worsens an agent’s future bargaining position. Proof ◮ If agent 1 wins in t , efficient bargaining means b t − 1 = 1 − U 1 ( k t 1 − b t , k t 2 , t − 1) − U 2 ( k t 1 − b t , k t 2 , t − 1) . ◮ At time t , b t given by indifference eqn (*), − b t = U 1 ( k t 1 , k t 2 − b t , t − 1) + U 2 ( k t 1 − b t , k t 2 , t − 1) − 1 . ◮ Summing these, b t − 1 − b t ≥ U 1 ( k t 1 , k t 2 − b t , t − 1) − U 1 ( k t 1 − b t , k t 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 b T = O (1 /T ) . Hence, U 1 ( k 1 , k 2 , T ) + U 2 ( k 1 , k 2 , T ) → 1 . Idea ◮ Bids escalate over time but sum to less than k 1 + k 2 . Rent dissipation fails ◮ Posner: The cost of obtaining a monopoly equals the profit of being a monopolist. ◮ If ( k 1 , k 2 ) ≥ (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 k 1 > 0 then ∃ T ∗ such that U 1 ( k 1 , k 2 , T ) > 0 for T ≥ T ∗ . Idea ◮ If agent 1 is held to zero utility then b t = k t 1 /t . ◮ This can be seen by induction, using indifference eqn (*) b t = k t 1 − b t b t = k t 1 ⇒ t − 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 ( k 1 , k 2 ) = (1 , 1) ◮ Shows tradeoff between agenda control and saving resources. ◮ Solve for utilities in closed form. ◮ Alternating offers is an equilibrium result. Defining ( k 2 1 , k 2 2 ) ∈ A t ◮ A 2 defined s.t. if 1 wins in t = 2 then k 1 1 ≤ min { k 1 2 , 1 } . ◮ A t defined s.t. ( k t − 1 , k t − 1 ) ∈ A t − 1 under the equilibrium bid. 1 2
Introduction Model Analysis Alternating The End Equilibrium Bids and Utilities Suppose ( k t 1 , k t Proposition 6. 2 ) ∈ A t for t ≥ 2 . Then bids are 2 ) = k t 1 + k t 2 − 1 b t ( k t 1 , k t , 2 t − 1 and agent 1 ’s continuation utility is 1 , t ) = 2 t − 1 − 1 1 − 2 t − 1 2 + 2 t − 1 U 1 ( k t 1 , k t 2 t − 1 k t 2 t − 1 k t 2 t − 1 . Implications ◮ Isobid property: transferring ǫ from 1 to 2 does not affect bids. ◮ Transfer property: transferring ǫ from 1 to 2 raises U 2 by ǫ . ◮ Doubling property: bids double each period.
Introduction Model Analysis Alternating The End Why Doubling? Intuition ◮ Winning costs b t directly and transfers b t to opponent. ◮ Winning enables agent to capture tomorrow’s loss, b t − 1 . Proof ◮ If agent 1 wins, the transfer property implies 2 , t − 1) + b t − 1 − b t U win ( k t 1 , k t 2 , t ) = U 1 ( k t 1 − b t , k t 1 2 − b t , t − 1) + b t − 1 − 2 b t . = U 1 ( k t 1 , k t ◮ If agent 1 loses, U lose ( k t 1 , k t 2 , t ) = U 1 ( k 1 , k 2 − b t , t − 1) . 1 ◮ Indifference and the isobid property implies 2 b t = b t − 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 ( k T 1 , k T 2 ) = (1 , 1) then, U 1 (1 , 1 , T ) = 2 T − 1 − 1 2 T − t b t = and 2 T − 1 . 2 T − 1 ◮ As T → ∞ then, U 1 ( k 1 , k 2 , ∞ ) = 1 2( k 1 − k 2 ) + 1 b T = O (1 / 2 T ) and 2 . ◮ Formally A t is characterized by � 2 t k t 2 t − 2 � 2 − 1 k t 3 · 2 t − 2 − 1 k t 1 ≤ min 2 t − 2 , 2 + 1 .
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