Bargaining Games An Application of Sequential Move Games The Bargaining Problem The Bargaining Problem arises in economic situations where there are gains from trade for example when a buyer values an item more than a seller The problem is how to divide the gains for example what price should be charged Bargaining problems arise when the size of the market is small and there are no obvious price standards because the good is unique e g a house at a particular location A custom contract to develop a web page etc We can describe bargaining games in extensive form that allow us to better understand the bargaining problem in various economic settings Bargaining Games A bargaining game is one in which two or more players bargain over how to divide the gains from trade The gains from trade are represented by a sum of money M that is on the table Players move sequentially making alternating offers Examples A Seller and a Buyer bargain over the price of a house A Labor Union and Firm bargain over wages benefits Two countries e g the U S and Canada bargain over the terms of a trade agreement The Disagreement Value If both players in a 2 player bargaining game disagree as to how to divide the sum of money M and walk away from the game then each receives their disagreement value Let a the disagreement value to the first player and let b the disagreement value to the second player In many cases a b 0 e g if a movie star and film company cannot come to terms the movie star doesn t get the work and the film company doesn t get the movie star The disagreement value is know by some other terms e g the best alternative to negotiated agreement BATNA By gains from trade we mean that M a b Take it or Leave it Bargaining Games Take it or leave it is the simplest sequential move bargaining game between two players each player makes one move Player 1 moves first and proposes a division of M For example x for player 1 and M x for player 2 Player 2 moves second and must decide whether to accept or reject Player 1 s proposal If Player 2 accepts the proposal is implemented Player 1 gets x and Player 2 gets M x If Player 2 rejects then both players receive their disagreement values a for Player 1 and b for Player 2 This game has a simple rollback equilibrium Player 2 accepts if M x b her disagreement value Used Car Example Buy me Buyer is willing to pay a maximum price of 8 500 Seller will not sell for a price less than 8 000 M 8 500 8000 500 a b 0 Suppose the seller moves first and there is perfect information the seller knows the maximum value the buyer attaches to the car Then the seller knows the buyer will reject any price p 8 500 and will accept any price p 8 500 The seller maximizes his profits by proposing p 8 500 or x 500 The buyer accepts since M x b The seller gets the entire amount M 500 What happens if the buyer moves first Ultimatum Game Discrete Version of Take it or Leave it Bargaining Player 1 moves first and proposes a division of 1 00 Suppose there are just 3 possible discrete divisions limited to 0 25 increments Player 1 can propose x 0 25 x 0 50 or x 0 75 for himself with the remainder 1 x going to Player 2 Player 2 moves second and can either accept or reject Player 1 s proposal If Player 2 accepts the proposal is implemented If Player 2 rejects both players get 0 each The 1 00 gains from trade vanish Problems with Take it or Leave it Take it or leave it games are too trivial there is no back and forth bargaining Another problem is the credibility of take it or leave it proposals If player 2 rejects player 1 s offer is it really believable that both players walk away even though there are potential gains from trade Or do they continue bargaining Recall that M a b What about fairness Is it really likely that Player 1 will keep as much of M as possible for himself The Dictator Game Are Player 1 s concerned about fairness or are they concerned that Player 2 s will reject their proposals The Dictator Game gets at this issue The Alternating Offers Model of Bargaining A sequential move game where players have perfect information at each move Players take turns making alternating offers with one offer per round i e this is real back andforth bargaining Round numbers t 1 2 3 Let 0 x t 1 be the fraction of M that player 1 asks for in bargaining round t and let 0 y t 1 be the fraction of M that player 2 asks for in bargaining round t Alternating Offer Rules Player 1 begins in the first round by proposing to keep x 1 M for himself and giving Player 2 1 x 1 M If Player 2 accepts the deal is struck If Player 2 rejects another bargaining round may be played In round 2 player 2 proposes to keep y 2 M for herself and giving 1 y 2 M to player 1 If Player 1 accepts the deal is struck otherwise it is round 3 and Player 1 gets to make another proposal Bargaining continues in this manner until a deal is struck or no agreement is reached an impasse is declared by one player a holdout If no agreement is reached Player 1 earns a and Player 2 earns b the disagreement values Alternating Offers in Extensive Form Round 1 Player1 Impasse x 1 M 1 x 1 M Player 2 a b Reject Accept Player 2 Impasse x 1 M 1 x 1 M 1 y 2 M y 2 M a b Player1 Reject Player1 Impasse a b Round 2 Accept 1 y 2 M y 2 M x 3 M 1 x 3 M Player 2 Round 3 Etc When Does it End Alternating offer bargaining games could continue indefinitely In reality they do not Why not Both sides have agreed to a deadline in advance or M 0 at a certain date The gains from trade M diminish in value over time and may fall below a b The players are impatient time is money Take it or leave it has 1 round deadline Let s focus on the last two possibilities Decreasing Gains from Trade Suppose there are 2 rounds Player 1 proposes a division of M first player 2 accepts rejects If accepted the proposal is implemented If rejected player 2 gets to make a counterproposal for how to split a reduced amount of money M where 0 1 is the rate of decay Player 1 can then accept or reject this final proposal The bargaining game is then over with certainty Example Suppose that M 12 1 3 Shrinking Gains …