Rubinstein negotiation model

The Rubinstein negotiation model (also Rubinstein negotiation solution ) is a model of non-cooperative negotiations discussed by Ariel Rubinstein (1982). This model belongs to game theory or negotiation theory .

The starting point is a situation where two individuals can make a number of possible contractual arrangements . Both have an interest in an agreement being reached, but their preferences are not completely identical. Assuming the theory of the rational decision that both act rationally, a proposed solution or the contract is to be found that both individuals would accept.

The model

Two players want to share a size 1 cake. The condition for this is that they agree on the shares that each should get. Player 1 starts and makes a bid for size . Hence, Player 2 would get. If player 2 accepts this proposal, the game ends and the cake is split into two as suggested by player 1 . If, however, player 2 does not accept the offer from player 1, the next negotiation period begins and player 2 makes an offer . In other words, is the share that player 1 would get and is the share that player 2 would get. If player 1 accepts the offer, the game is over and the cake is divided into. However, if player 1 does not accept the offer proposed by player 2, a third negotiation period begins and player 1 will make another offer. The negotiation process can potentially take an infinite amount of time. ${\displaystyle {\begin{aligned}x\in [0,1]\end{aligned}}}$ ${\displaystyle {\begin{aligned}(1-x)\end{aligned}}}$ ${\displaystyle {\begin{aligned}(x,1-x)\end{aligned}}}$ ${\displaystyle {\begin{aligned}y\in [0,1]\end{aligned}}}$ ${\displaystyle {\begin{aligned}y\end{aligned}}}$ ${\displaystyle {\begin{aligned}(1-y)\end{aligned}}}$ ${\displaystyle {\begin{aligned}(y,1-y)\end{aligned}}}$

The negotiation is stationary , as both players have to decide in each round which offer to propose and whether to accept the offer. The offer of a round is independent of the offers from previous rounds. Since the players do not learn from the past (assuming complete information ), the negotiation game will always look the same in the future.

Formal representation

Successful negotiation results are generally represented by tuples . Here is Player 1's share of the total, regardless of which player made this suggestion. It is believed that the players the contracts from the negotiations result given their preference relation over can arrange. The utility functions of the players are in the 1st round or in the 2nd round. ${\displaystyle (s,t),s\in [0,1],t\in \mathbb {N} _{0}}$ ${\displaystyle {\begin{aligned}s\end{aligned}}}$ ${\displaystyle {\begin{aligned}\succsim _{i}(i=1,2)\end{aligned}}}$ ${\displaystyle {\begin{aligned}([0,1])\times \mathbb {N} _{0}\end{aligned}}}$ ${\displaystyle {\begin{aligned}(U_{1},U_{2})=(s_{1},(1-s_{1}))\end{aligned}}}$ ${\displaystyle {\begin{aligned}(U_{1},U_{2})=(s_{2},1-s_{2})\end{aligned}}}$

The game tree

Game tree of the Rubinstein game (The game tree shows the course of a negotiation. The players take turns submitting bids and not at the same time considering the acceptance of the complete information.)

The extensive form of this negotiation game is mapped with the help of a game tree. Each negotiation period consists of two stages, with the first stage showing a split proposal that the opponent can accept ("Y") or reject ("N") at the next stage. The end point of a game tree is denoted by a tuple , or . This not only shows the overall breakdown or distribution resulting from the contract , but also the end time for the respective negotiation process. If you want to describe a game completely, you have to define the respective utility functions of the players on the end points of the game tree. ${\displaystyle {\begin{aligned}(x,t)\end{aligned}}}$ ${\displaystyle {\begin{aligned}(y,t)\end{aligned}}}$ ${\displaystyle {\begin{aligned}(x,1-x)\end{aligned}}}$ ${\displaystyle {\begin{aligned}(y,1-y)\end{aligned}}}$ ${\displaystyle {\begin{aligned}t\end{aligned}}}$

Assumptions in the model

Players can use the results of a hearing in accordance with their preference relation over rate and assign. The following assumptions apply to player preferences: ${\displaystyle {\begin{aligned}(i=1,2)\end{aligned}}}$ ${\displaystyle {\begin{aligned}([0,1])\times \mathbb {N} _{0}\end{aligned}}}$

Assumption of monotony

Player 1 always prefers a contract that guarantees him a share of to a division of , as long as he secures the larger share  ; ${\displaystyle {\begin{aligned}s\end{aligned}}}$ ${\displaystyle {\begin{aligned}s'\end{aligned}}}$ ${\displaystyle {\begin{aligned}s\end{aligned}}}$ ${\displaystyle {\begin{aligned}\to (s>s')\end{aligned}}}$

Impatience acceptance

Future division is worth less than the present one;

Continuity assumption

Small changes in the contract about the division between the players cause just as small changes in the preference for such a contract;

Stationarity assumption

The structure of a negotiation remains constant in every period, i.e. regardless of the bids from previous periods, one player must submit a bid and the other player can accept or reject it (players do not learn from the past).

Complete information

All players know their time preference and that of their opponent.

Impatience in the model

The central assumption is that the players not only rate their share of the cake (or for player 2), but also rate the point in time at which they receive the piece of cake. This assumption is called impatience . This means that a payout is worth more to a player at the time than it is at the time . In a subgame-perfect equilibrium , the negotiation cannot have an infinite number of periods, otherwise payouts of zero would result. Impatience can be by means of discounting, or the discount rate , represent: ${\displaystyle {\begin{aligned}x\end{aligned}}}$ ${\displaystyle {\begin{aligned}(1-x)\end{aligned}}}$ ${\displaystyle {\begin{aligned}t\end{aligned}}}$ ${\displaystyle {\begin{aligned}t\end{aligned}}}$ ${\displaystyle {\begin{aligned}(t+1)\end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta ={\frac {1}{1+r}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}r={\text{Zinssatz}}\end{aligned}}}$

• ${\displaystyle {\begin{aligned}r=0\end{aligned}}}$and , negotiation results are not discounted and the players are extremely patient ${\displaystyle {\begin{aligned}\delta =1\end{aligned}}}$
• ${\displaystyle {\begin{aligned}r=\uparrow \end{aligned}}}$and , bargaining results are discounted with each period and the players are impatient ${\displaystyle {\begin{aligned}\delta =\downarrow \end{aligned}}}$
• ${\displaystyle {\begin{aligned}r\to \infty \end{aligned}}}$and , players are extremely impatient and this means that negotiation results in the 2nd round are no longer worth anything. ${\displaystyle {\begin{aligned}\delta \to 0\end{aligned}}}$

criticism

The assumption of stationarity is contradicted with regard to the impatience of players. Waiting for tomorrow seems worse to individuals compared to increasing the waiting time from 100 days to 101 days.

Complete information describes a state in which the players can observe time preferences and preferences about the division of the cake. However, in reality it is difficult to determine or observe. There is a form of impatience in negotiations and possible extreme cases as exist, but if the opponent were selfish he would not admit this, since any form of threats on his part would be implausible. ${\displaystyle {\begin{aligned}\delta \end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta =0\end{aligned}}}$

Equilibrium solution

Nash equilibrium

The contract is a Nash equilibrium when neither player has any incentives to deviate from the equilibrium dictate of what the opponent does. ${\displaystyle {\begin{aligned}(x^{*},(y^{*}=1-x^{*}))\end{aligned}}}$

procedure

The Nash equilibrium achieved in a negotiation consists of two parts:

1. When it is player 1's turn (in the odd periods it does) he will always propose. Player 2 will reject each if , because then is . So player 2 will assume if . ${\displaystyle {\begin{aligned}x^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}x\end{aligned}}}$ ${\displaystyle {\begin{aligned}x>x^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}1-x<1-x^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}x\end{aligned}}}$ ${\displaystyle {\begin{aligned}x\leq x^{*}\end{aligned}}}$
2. When it is player 2's turn (in the even periods) he will always propose. Player 1 rejects all bids if . But if true, player 1 will accept. ${\displaystyle {\begin{aligned}y^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}y\end{aligned}}}$ ${\displaystyle {\begin{aligned}y>y^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}y\leq y^{*}\end{aligned}}}$

interpretation

If player 2 is in situation 1 and rejects player 1's offer, he assumes that he can enforce the offer that is more advantageous for him in situation 2 . Player 1 cannot offer any other offer than that, since he assumes that player 2 will behave given situation 1. Given a suitable one , there is a Nash equilibrium for each . ${\displaystyle {\begin{aligned}x^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}y^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}x^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}y^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}x^{*}\end{aligned}}}$

First mover advantage

First-Mover-Advantage describes the possibility of determining the starting position of the subsequent game in which a player begins.

Assume that and are not specified. If the negotiation begins with an odd period, i.e. player 1 makes the first bid, the Nash equilibrium is reached, given player 2 behaves as assumed in situation 1. ${\displaystyle {\begin{aligned}x^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}y^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}(x^{*},1-x^{*})\end{aligned}}}$

If player 2 starts, he will offer that player 2 is assuming situation 2. If the assumption is valid for the behavior of player 1 given situation 2, then player 2 can make a bid that is very close to 1 and player 1 would not refuse. ${\displaystyle {\begin{aligned}y^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}y^{*}\end{aligned}}}$

Result

In both situations 1 and 2, strategies that are expressed by the payouts and represent a Nash equilibrium . However, this does not imply that the equilibrium found is also subgame perfect . Both player 1 and player 2 have the option of deviating from the equilibrium strategy in both situations. The possibility of deviating represents a threat , as both have an incentive to enforce their offer. ${\displaystyle {\begin{aligned}x^{*}\end{aligned}}}$ ${\displaystyle {\begin{aligned}y^{*}\end{aligned}}}$

Partial perfect balance

In order for the found balance to be subgame perfect, a player must be indifferent between the payout that he receives at the time he accepts the offer and the payout that he would receive if he were to reject the offer . This must apply to both players in each period. Part-game perfection implies that player 2 is also taken into account as player with his bids in the 2nd period , as this represents a new part-game. ${\displaystyle {\begin{aligned}t=0\end{aligned}}}$ ${\displaystyle {\begin{aligned}t=1\end{aligned}}}$ ${\displaystyle {\begin{aligned}t=0\end{aligned}}}$ ${\displaystyle {\begin{aligned}t=1\end{aligned}}}$

Backward induction

Assumptions

It is assumed that the utility function of both players includes time as a restriction in the function with the help of discount factors. It is described as: ${\displaystyle {\begin{aligned}v\end{aligned}}}$ ${\displaystyle {\begin{aligned}v_{i}\end{aligned}}}$

• ${\displaystyle {\begin{aligned}v_{i}=\delta _{i}^{t}s_{i}\end{aligned}}}$for for players . ${\displaystyle {\begin{aligned}0\leq \delta \leq 1\end{aligned}}}$ ${\displaystyle {\begin{aligned}i=1,2\end{aligned}}}$

solution

Part-game-perfect contract (Here: graphical solution by considering the indifference relationships of both players. The intersection of the utility functions shows the sub-game-perfect contract of both players.)

Solution by backward induction : It must apply to the players that they are indifferent between the offer from the opponent and their own proposal (given who is the "first mover" of the negotiation). ${\displaystyle {\begin{aligned}t=0\end{aligned}}}$ ${\displaystyle {\begin{aligned}t=1\end{aligned}}}$

Player 1	Player 2
${\displaystyle {\begin{aligned}\delta _{1}^{t}s_{1}=\delta _{1}^{t+1}s_{2}\end{aligned}}}$	${\displaystyle {\begin{aligned}\delta _{2}^{t}(1-s_{1})=\delta _{2}^{t+1}(1-s_{2})\end{aligned}}}$
or. ${\displaystyle {\begin{aligned}s_{1}=\delta _{1}s_{2}\end{aligned}}}$	${\displaystyle {\begin{aligned}(1-s_{1})=\delta _{2}(1-s_{2})\end{aligned}}}$

With the help of the equation, the equilibrium values ​​can be determined for discount rates of the same size (here player 1 "first mover"): ${\displaystyle {\begin{aligned}(s^{*},(1-s^{*})\end{aligned}}}$ ${\displaystyle {\begin{aligned}(\delta _{1}=\delta _{2}=\delta )\end{aligned}}}$

• ${\displaystyle {\begin{aligned}s^{*}=s_{1}={\frac {1-\delta }{1-\delta ^{2}}}={\frac {1}{1+\delta }}\end{aligned}}}$

If the players have different levels of patience, the equal contract is (here for player 1):

• ${\displaystyle {\begin{aligned}s^{*}=s_{1}={\frac {1-\delta _{2}}{1-\delta _{1}\delta _{2}}}\end{aligned}}}$

interpretation

Pressure to reach an agreement (the share of the pie shrinks the longer the negotiation. The graph shows, given the discount rates of both players, how the shares of the players in the pie are falling continuously.)

Patience plays a big role in determining what balance will be achieved and how long a negotiation can potentially take. Both players have different levels of patience, i.e. they have different preferences over time. So the player who has a higher discount rate and is therefore more patient is preferred.

The following examples illustrate the effect of discount factors on the negotiation:

example 1

If player 1 starts an offer , he can assume that player 2 will accept anything at a discount rate of (player 2 is extremely impatient), since player 2 is no longer worth an allocation of the cake in the next period. Player 1 can even secure the whole cake for himself, since player 2 is indifferent between the state in the 1st period in which he does not receive a share and the next in which his share is no longer worth anything to him. to decline is an empty threat from player 2. Although player 2 does not get a share of the cake, the sub-game-perfect balance is achieved. This again illustrates the advantage of being a "first mover", because even if player 1 were impatient , he could demand the whole cake from an extremely impatient opponent and player 2 would accept it. ${\displaystyle {\begin{aligned}s_{1}\end{aligned}}}$ ${\displaystyle {\begin{aligned}s_{1}\end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta _{2}=0\end{aligned}}}$ ${\displaystyle {\begin{aligned}s_{1}\end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta _{1}\to 0\end{aligned}}}$

Example 2

If player 1 starts with a proposal , but is extremely impatient , he can still secure a share of (if ). ${\displaystyle {\begin{aligned}s_{1}\end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta _{1}=0\end{aligned}}}$ ${\displaystyle {\begin{aligned}1-\delta _{2}\end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta _{2}<1\end{aligned}}}$

Partial perfect balance: ${\displaystyle {\begin{aligned}s^{*}=s_{1}={\frac {1-\delta _{2}}{1-\delta _{1}\delta _{2}}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}\Leftrightarrow s^{*}=s_{1}=1-\delta _{2}\end{aligned}}}$

This also shows that it is an advantage to go into the negotiation as a "first mover". If player 2 started, the situation from example 1 would occur and player 2 would secure the entire cake.

Example 3

If both player 1 and player 2 have the same discount rates (it applies ) and it applies , the result of the negotiation approaches the equal distribution and this also corresponds to the Nash solution, since the game is symmetrical. That means both players get half the cake each, so . ${\displaystyle {\begin{aligned}\Leftrightarrow \delta _{1}=\delta _{2}=\delta \end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta \to 1\end{aligned}}}$ ${\displaystyle {\begin{aligned}(s^{*},(1-s^{*})=({\frac {1}{2}},{\frac {1}{2}})\end{aligned}}}$

If the discount rate for both players is included , it is worth starting the negotiation again as a "first mover", since if player 1 starts it would apply. Player 1 can therefore secure a larger share of the cake. ${\displaystyle {\begin{aligned}\delta <1\end{aligned}}}$ ${\displaystyle {\begin{aligned}(s^{*}>{\frac {1}{2}})\end{aligned}}}$

Risk of breaking off negotiations

Risk of termination of negotiations (The graphic shows the Rubinstein negotiation game in extensive form . The uncertainty among the players about a possible termination of the negotiation is represented with the help of probabilities . The first two periods are shown; the game can potentially last indefinitely.) ${\displaystyle {\begin{aligned}\alpha \end{aligned}}}$

Impatient gamblers prefer short negotiations, as they assume that the cake will be worthless in the future. There is a possibility that a player might lose interest in the course of a negotiation and the negotiation unexpectedly ends. This termination probability is expressed by. So there is a probability that the negotiation will continue. The risk of abandonment puts a high pressure on both players to agree, so . If the negotiation breaks off prematurely, both players receive a share of , where (the same applies to in the 2nd round), or . If an agreement is reached in the 1st round, player 1 receives a payout from or player 2 . The balanced payout no longer depends only on the patience of the players, but on the possibilities that develop outside of the negotiation and lead to a negotiation being abandoned. ${\displaystyle {\begin{aligned}1-\alpha \end{aligned}}}$ ${\displaystyle {\begin{aligned}\alpha \end{aligned}}}$ ${\displaystyle {\begin{aligned}\delta _{1}=\delta _{2}=\delta \to 1\end{aligned}}}$ ${\displaystyle {\begin{aligned}(b_{1},b_{2})\end{aligned}}}$ ${\displaystyle {\begin{aligned}0<b_{1}<x\end{aligned}}}$ ${\displaystyle {\begin{aligned}y\end{aligned}}}$ ${\displaystyle {\begin{aligned}0<b_{2}<1-x\end{aligned}}}$ ${\displaystyle {\begin{aligned}x\end{aligned}}}$ ${\displaystyle {\begin{aligned}1-x\end{aligned}}}$

literature

• Ariel Rubinstein: Perfect equilibrium in a bargaining model. In: Econometrica. Volume 50, No. 1, 1982, pp. 97-110.
• Manfred J. Holler, Gerhard Illing: Introduction to game theory. Springer, 2005, ISBN 3-540-27880-X .
• SK Berninghaus, K.-M. Ehrhart, W. Güth: Strategic games. Springer, 2005, ISBN 3-540-28414-1 .
• Martin Osborne, Ariel Rubinstein: A Course in Game Theory. MIT Press, 1994.
• Robert Gibbons: A Primer in Game Theory. Pearson, 1992.
• David Laibson: Golden Eggs and Hyperbolic Discounting. Oxford University Press, 1997.
• MJ Holler: Economic Theory of Negotiations. Oldenburg Verlag, 1992.
• Martin J. Osborne: An Introduction to Game Theory. Oxford University Press, 2004.

Individual evidence

1. ^ A. Rubinstein: Perfect Equilibrium in a Bargaining Model. In: Econometrica. Volume 50, No. 1, 1982.
2. ↑ SK Berninghaus, K.-M. Ehrhart, W. Güth: Strategic games. 2nd Edition. Springer, 2005, ISBN 3-540-28414-1 .
3. ↑ David Laibson : Golden Eggs and Hyperbolic Discounting. In: The Quarterly Journal of Economics. Vol. 112, Oxford University Press, 1997.
4. ↑ Manfred J. Holler, Gerhard Illing: Introduction to game theory. Springer, 2005.
5. ^ Martin J. Osborne: An Introduction to Game Theory. Oxford University Press, 2004.