The Rubinstein bargaining game without an exogenous first-mover

Transcription

1 The Rubinstein bargaining game without an exogenous first-mover Fernando Branco Universidade Católica Portuguesa First Version: June 2007 This Version: January 2008 Abstract I study the equilibria of a modified Rubinstein alternating offer bargaining game, in which no player is exogenously given a role of first mover Concentrating the analysis in mixed strategy equilibria, I show that delays appear in equilibrium with positive probability If players have similar and low discount factors, the competition is very intense In the limit, with myopic players, they propose in every period and agreement is never reached On the other hand, if discount factors are very different, the more patient player proposes with a higher probability but agreement is faster Anyway, the expected moment of agreement is always above 2, and may be quite large JEL Classification: C72, C78 Please address correspondence to Fernando Branco, Universidade Católica Portuguesa, FCEE, Palma de Cima, Lisboa, Portugal (fbranco@ucppt) A previous version of this paper was presented at the 2007 ASSET Meeting in Padova A grant from Fundação para a Ciência e Tecnologia (POCTI/ECO/13133/98/2001) is gratefully acknowledged

2 1 Introduction In his seminal paper, Rubinstein (1982) considered a dynamic model of bargaining between two players, which alternate in proposing the division of a shrinking pie 1 Quite surprisingly, in Rubinstein s own words, the game has a unique subgame perfect equilibrium The player that starts the bargaining (labelled as player 1) obtains the fraction (1 δ 2 )/(1 δ 1 δ 2 ) of the pie, where δ i is player i s discount factor Hence, the equilibrium division of the pie depends only on the players discount factors and on who starts the bargaining In this note, I study the equilibria of a modified Rubinstein alternating offer bargaining game, in which no player is exogenously given a role of first mover The modified game proceeds as follows In the first period, both players simultaneously select an action, which may either be to propose a division of the pie or the decision of no proposal If one player only proposes a division in the first period, the game will proceed as in the Rubinstein bargaining game: the other player may accept or reject; if he accepts, the division takes place and payoffs are realized; if he rejects, it will be his turn to propose a division in the following period; and so on If neither player or both players propose divisions in the first period, the game advances to the second period, where both players simultaneously select an action (either proposing a division or not proposing) and the game proceeds as described for the previous period I find an interest in this modified Rubinstein bargaining game for several reasons First, the players start the game in symmetric positions relative to the order of moves Second, the players are allowed not only to propose divisions but also to decide on when to make a proposal Third, the model allows for studying the role of discount factors not only on the proposed divisions but also on the decision to propose and the time of agreement Fourth, there are equilibria in which delays happen with positive probability In the following section I develop the analysis of the model First, I provide some general properties of the equilibrium strategies Then, I proceed with the analysis of two cases: I start with the case in which both players have a common discount factor; then, I continue with the case of distinct discount factors 1 In this paper I refer only to the fixed discounting factor sub-family of the model 1

3 2 The analysis of the model There may be many equilibria in the modified Rubinstein bargaining game For example, two of the equilibria correspond to the equilibria in the Rubinstein bargaining game with either player as the first-mover: in a period in which both players can propose, one of the players always proposes and the other player never does it In this note, however, I concentrate the analysis to equilibria in which both players propose divisions with positive probability in every period in which both players have the opportunity to propose To be more precise, I explore equilibria of the following form: In period 1, with probability α i player i proposes x i for him and 1 x i for the other player, and with probability 1 α i player i does not present a proposal; If a single player has proposed in period 1, the game will proceed as in the Rubinstein alternating offer bargaining game; Otherwise, the game moves to the period 2, and with probability α i player i proposes x i for him and (1 x i ) for the other player, and with probability 1 α i player i does not present a proposal; The game continues as in the previous period I first characterization of the equilibrium strategies is provided in the following proposition Proposition 1 In any equilibrium, within the class analyzed, the divisions proposed by player i are the same as the divisions proposed in the equilibrium of the Rubinstein alternating offer bargaining game I find useful to think about this result in the following way In the beginning of a period in which a player is allowed to propose a division, he has a two step decision: first, he needs to decide whether or not he wants to propose a division; second, if proposing, he needs to decide which division to propose But, his proposal in a given period will only be payoff relevant if it is the unique proposal, and from that moment on the game follows as the Rubinstein alternating offer bargaining game So, in this second step, the player faces exactly the same trade-offs he faces in 2

4 the Rubinstein alternating offer bargaining game Therefore, in equilibrium, the divisions proposed are the ones proposed in the equilibrium of the Rubinstein alternating offer bargaining game I continue the analysis by separating the case in which the players have a common discount factor and that in which they have different discount factors 21 The common discount factor case I consider first the case in which both players have the same discount factor, δ Following Proposition 1, in any equilibrium, whenever a player proposes a division, he proposes a share x = 1/(1+δ) of the pie for himself, leaving the remaining share 1 x = δ/(1 + δ) for the other player If he is the unique player proposing a division in that period, such a division will immediately be accepted by the other player I will look at mixed strategy equilibria, in which, in all periods when both players are allowed to propose a division, player i will make an offer with probability α i (0, 1) Proposition 2 There is a unique equilibrium in (strictly) mixed strategies: in any period in which both players can propose a division, each player proposes a share x = 1/(1 + δ) of the pie for himself, with probability α = 1 + 4δ 1 2δ The first issue to remark is that the players will propose a division with the same probability, ie, there are no (strictly) mixed strategy equilibria in which the players use different probabilities In this equilibrium each player proposes a division with a probability that varies monotonically α δ Figure 1: Probability of proposing a division 3

5 between 1 and ( 5 1)/2, as δ goes from 0 to 1 (see Figure 1) Hence, the more patient the players are, the lower is the probability that any player proposes a division The equilibrium expected payoff of each player, π is given by: π = 1 + 2δ 1 + 4δ ( δ(1 + δ) 3 ), 1 + 4δ which monotonically varies between 0 and 05, as δ goes from 0 to 1 (see Figure 2) π δ Figure 2: Expected payoff It is also possible to compute the probability of agreement in the current period, conditional on having reached that period, β, which is: β = 2α (1 α ) = (1 + δ) 1 + 4δ (1 + 3δ) δ 2 This probability varies monotonically between 0 and 2 5 4, as δ goes from 0 to 1 Hence, the probability of agreement in the current period, conditional on not having an agreement before, is greater as the players get more patient, but it is always below 05 (see Figure 3) β Figure 3: Probability of agreement in the current period δ 4

6 Therefore, delays in reaching an agreement are highly probable Indeed the period of agreement is a random variable with a geometric distribution Some cases, for several values of δ, are depicted in Figure 4 P ro b a b i l i t y o f a g re e m en t Period Figure 4: Probability of agreement in each period ( δ = 01; δ = 05; δ = 09) Agreement is faster for more patient players This is nicely captured by the determination of the expected moment of agreement: E {T } = 1 β = δ 2 (1 + δ) 1 + 4δ (1 + 3δ), which monotonically varies from + to 1/(2 5 4), as δ goes from 0 to 1 (see Figure 5) Hence, the expected agreement moment is lowered as the discount factor increases (more patient players), but it is always above two periods E {T } δ Figure 5: Expected agreement period The results in this subsection can be read in terms of the intensity of competition between 5

7 the two players In one extreme case, if the players are myopic (δ = 0), they will be driven to a Bertrand-like competition: both players will propose a division in every moment; there is never an agreement; and their payoffs are zero As the players become more patient the competition becomes less intense Nevertheless, even for infinitely patient players, their are delays in the agreement, which comes on average after moment 2, which however does not affect the expected payoff, which is equal to The distinct commonly known discount factor case Suppose now that the two players have distinct discount factors (δ i is player s i discount factor) From Proposition 1, we know that, when proposing a division, player i will ask for a share x i = (1 δ j )/(1 δ i δ j ) of the pie for himself, leaving the remaining share 1 x i = δ j(1 δ i )/(1 δ i δ j ) for player j I now look for a mixed strategy equilibrium, in which in periods when both players are allowed to propose a division, player i will propose with probability α i characterizes such equilibrium (0, 1) The following Proposition Proposition 3 There is a unique equilibrium in (strictly) mixed strategies: in any period in which both players can propose a division, player i proposes a share x i = (1 δ j)/(1 δ i δ j ) of the pie for himself, with probability α i = 1 + 4δj 1 2δ j The expression for α i factor analyzed in Subsection 21 is a generalization of the expression obtained for the common discount The probability that a player proposes a division in the current period, depends on the other player s discount factor: the more patient the other player is the lower the probability of proposing a division Hence, the most patient player will be more likely to propose a division, ie, if δ 1 > δ 2, then α1 > α 2 (see Figure 6) The expected equilibrium payoff of player i is given by: π i = (1 δ j)(1 + 2δ i 1 + 4δ i ) δ i (1 δ i δ j )( δ i ), 6

8 α i α 1 with δ 2 = 01 α 1 with δ 2 = 025 α 1 with δ 2 = 05 α 1 with δ 2 = 075 α 2 δ 1 Figure 6: Probability of proposing a division which was derived in the proof of Proposition 3 The expected payoff increases in the player s own discount factor and decreases in the other player s discount factor In fact: Given δ 2 1, πi varies continuously from 0 to 1 as δ 1 goes from 0 to 1 2 Given δ i, πi varies from (1 + 2δ i 1 + 4δ i )/(δ i ( δ i )) to 0 as δ 2 goes from 0 to 1 δ π1 = 01 π1 = 02 π1 = 035 π1 = δ 1 Figure 7: Expected payoff of player i Hence, the most patient player will get a higher expected payoff The evolution of π 1 and π 2 for some values of the parameters is depicted in Figure There is a discontinuity if δ 2 = 1, in which case π i = 0, except that π i = 05 if δ 1 = 1 3 The dashed curves are π 2 s iso-level curves for the values 01, 02, 035 and 055 Note that for the same value π 1 and π 2 the iso-level curves intersect on the diagonal 7

9 It is also possible to compute the probability of agreement in the current period, conditional on having reached that period, β, which is: β = α 1(1 α 2) + α 2(1 α 1) = 1 + 4δ δ δ δ δ 2 2δ 1 2δ δ δ2 1 2δ 2 β δj = 09 δ j = 05 δ j = 01 Figure 8: Probability of agreement in the current period This probability varies monotonically between 0 and 2 5 4, as δ goes from 0 to 1 Its evolution is depicted in Figure 8, for several values of the discount factors The probability of an agreement in the current period increases in both discount factors Hence, when either player becomes more patient the probability of agreement in the current period increases δ i The expected moment of agreement decreases in both discount factors player becomes more patient the agreement is faster in expectation 120 Hence, when either E {T } δ j = 01 δ j = δ j = Figure 9: Expected agreement moment δ i 8

10 3 Conclusion In this note, I studied equilibria of a modified Rubinstein alternating offer bargaining game, in which no player is exogenously given a role of first mover Hence, right from the first period, any player may decide to propose a division of the pie While none or both players propose a division, there is no agreement and the game proceeds to the following period with a similar round of possible proposals When, in a given period, a single player proposes a division, the game continues as in the Rubinstein bargaining game I concentrated the analysis of the model in mixed strategy equilibria in which players propose a division with the same probability in any period in which both players are allowed to propose a division In equilibrium, whenever allowed, a player proposes a division with strictly positive probability Therefore, delays in reaching the agreement happen in equilibrium The expected moment of agreement is greater than 2, and may be quite large for small discount factors If both players are very patient, the probabilities that each will propose a division in the initial period are lower Hence, agreement is faster and payoffs are higher On the other extreme, if both players are (close to) myopic, the probabilities that each will propose a division grows to approach 1 Agreements are delayed and payoffs are lower In the limit, as players do not value the future, a Bertrand-like competition emerges: both players propose divisions in every period and there is never an agreement When the game is played by players with different discount values, the more patient player will be more likely to propose a division in the initial period and he earns a higher expected payoff There may exist other equilibria in the game The characterization of other equilibria will be the next step in this note All the analysis in this note has been done under complete information The next step in this research will be to analyze the model if players are privately informed about their discount factors Appendix: Proofs Proof of Proposition 1 The proof is straightforward 9

11 Proof of Proposition 2 Suppose that player 2 behaves in the assumed way If player 1 proposes a division in period 1, his expected payoff will be: (1 α2)x + α2δπ 1 If, instead, player 1 decides not to propose a division in period 1, his expected payoff will be: α 2(1 x ) + (1 α 2)δπ 1 Hence, player 1 will be indifferent between the two choices in period 1, if and only if: (1 α2)x + α2δπ 1 = π1 α2(1 x ) + (1 α2)δπ 1 = π1 Solving this system of equations in order to α 2 one obtains a unique solution in (0, 1): α 2 = 1 + 4δ 1 2δ Hence, if α 2 = ( 1 + 4δ 1)/(2δ), player 1 will be willing to randomize over whether to propose a division of not to The problem of player 2 is similar to the problem of player 1 Therefore, it can be concluded that the equilibrium is unique, and that each player while both players are allowed to propose a division, both will propose the share x = 1/(1 + δ) of the pie for himself, with probability α = 1+4δ 1 2δ Proof of Proposition 3 Denote by πi the equilibrium payoff of player i in such an equilibrium The equilibrium payoffs associated to each choice in period 1 are presented in the following table: 4 4 This table should not be confused with a payoff matrix of a strategic form game 10

12 Player 2 Player 1 Offer Offer δ 1 π 1, δ 2π 2 No offer No offer 1 δ 2, δ 2(1 δ 1 ) 1 δ 1 δ 2 1 δ 1 δ 2 δ 1 (1 δ 2 ) 1 δ 1, δ 1 π1 1 δ 1 δ 2 1 δ 1 δ, δ 2π2 2 The values of α j and π i can be determined jointly from the following conditions: αjδ i πi + (1 αj) 1 δ j = πi 1 δ i δ j αj δ i (1 δ j ) + (1 α 1 δ i δ j)δ i πi = πi j The conditions to determine α i and π j are similar Solving these conditions one obtains: 1 + αi 4δj 1 = 2δ j πi = (1 δ j)(1 + 2δ i 1 + 4δ i ) δ i (1 δ i δ j )( δ i ) References Rubinstein, A (1982) Perfect Equilibrium in a Bargaining Game, Econometrica, 50(1),