Emotions in bargaining

Table of contents

Table directory

Figure directory

Introduction
1. A case of negotiation
2. Two possible outcomes and emotions

A) Bargaining theory
1. An introduction into game theory
i. A simple game and dominant strategies
ii. The Nash equilibrium and the sub-game perfect equilibrium
iii. Pure and mixed strategies
2. Games in bargaining theory
i. Cooperative bargaining and the Nash bargaining solution
ii. The Rubinstein model: a non-cooperative game
a) Introduction and example
b) Equilibrium
c) Implementing alternative utilities
d) Uniqueness of equilibrium

B) Emotions in bargaining- theory – an overview
1. Incorporating fairness into game theory and economics – Matthew Rabin (1993)
2. Modeling altruism and spitefulness in experiments – David K. Levine (1998)
3. A theory of fairness, competition and cooperation
– Ernst Fehr and Klaus Schmidt (1999)
4. A theory of equity, reciprocity and competition
– Gary E. Bolton and Axel Ockenfels (2000)
5. Bargaining with reference dependent preferences –Compte and Jehiel (2003)
6. Introducing social norms in game theory – Raúl López-Pérez (2007)
7. Bargaining with history-dependent preferences – Duohze Li (2007)
8. Strategic extremism: bargaining with endogenous breakdown probabilities and order of play – Elie Appelbaum (2008)

C) How to bring emotions into bargaining
– feasible models of negotiation for a higher wage
1. Comparing two outcomes - a closer look
2. Modelling two outcomes
i. Possible outcome one: leaving in anger
ii. Possible outcome two: the terror regime
3. A general model for the emotional negotiation for wages
i. How a model for emotional bargaining could look like
ii. An alternative form

Conclusion

Literature

Table directory

Table 1: a simple game

Table 2: the simple market game

Table 3: modified simple game for mixed strategies

Table 4: lower and upper bounds for a unique equilibrium

Table 5: three turns of the Rubinstein-game with breakdown

Table 6: utilities in a three period example with

Table 7: solving the scenario “bargaining for wages” with emotions for equilibrium

Table 8: preferences: player A is more emotional than player B, barriers in color

Table 9: example for the model proposed

Table 10: example for the proposed, modified model

Figure directory

Figure 1: sequential game with two sub-games

Figure 2: Rubinstein-game in extensive form

Figure 3: Fehr and Schmidt: utility function

Figure 4: additively separable motivation function: c=1. a/b =1/4. = ¾, = 1/5.

Figure 5: penalty function +1, f= 0.5

Figure 6: (𝑥, 𝑓) -e^(x-f)+1, = x-e^(x-f)+1, f=0.5

Figure 7:

Introduction

In the field of economic theory emotions are not the central point. Economics and the behavior of individuals in economic sets is thought and taught to be dominated by numbers and trade-offs centering on a utility that is tradable and countable. It is true that our modern working life with its sophisticated means of division of labor is very abstract and the bookkeeping tells us the worth of every individual operation by a number. While the benefits of such a complicated system are unarguably observable by the high standard of living of the industrialized countries, for the worker him-or herself it is often hard to imagine that his or her work is of any social worth. If it is reasonably assumed that the dimension of anything is money, calculation can be done very comfortably. Usually, any model in economics assumes a rational behavior of all agents. But irrationality did also get some thought, and so did emotion to some degree. The sense of economics itself is to describe how mankind acts to satisfy its needs by producing and trading - and satisfaction is an emotional value. When doing research on how people purchase things and produce goods it is unarguable a good idea to keep models simple in that concern so that rationality and logical decisions are assumed. When doing research on a smaller scale or even on an individual level however, emotions, play a far more important role.

This thesis focuses on emotions in bargaining. Negotiation happens on a very individual level and the assumption that bargaining happens without emotion is not feasible. Therefore emotion already got some attention in bargaining theory. The center of interest in most researches is fairness. The outcomes of many bargaining experiments were driven by some kind of altruism which then researchers tried to reproduce by models (cf. Fehr and Schmitt 1999, Levine 1998). Other emotions did not get so much attention.

1. A case of negotiation

A typical bargaining situation that every person in our culture will face at least a few times in life is, when applying for a job or negotiating for wages. Suppose, as a first player in a bargaining game, a young adult who is well educated and is just applying for the first job. After being invited into the firm she faces the second player, an older colleague, who represents the firm but also has intrinsic motivation and goals for himself. Both players negotiate upon the wage of the young professional, assuming she agreed to do the job in general and the firm also accepts her as a new employee. She has certain material needs and wishes, but she also plans for the future. Thus, she wants a certain amount of money on top of the amount that would satisfy her very basic needs. She does so because she does not want to be reputed as somebody who agrees to low wages although doing professional work. She is afraid of not getting good jobs after her first one, because she believes that a relatively low wage represents relatively poor work. Moreover, she might not get very high wages in the future, when her material needs may be higher than now due to having a family on her own etc. On the other hand her soon-to-be boss has his department in mind: high wages will stress his department's budget and lower his bonus. He is also afraid of being reputed as somebody who pays high wages and therefore will risk a worse stand in future bargaining. What are the possible outcomes of such a game considering the emotions of both players?

2. Two possible outcomes and emotions

Maybe both players will agree on an acceptable compromise and will live on happily ever after. This is surely true for many real negotiations. But this is not what this paper is about. Surely a relatively happy outcome can very well be described by a lot of means of game theory and economic models and emotions do not even necessarily have to be considered in detail to achieve this outcome. Any outcome where the agents maximized their utility until they found equilibrium can be considered as an outcome of this kind. This paper deals with the possible upsides and downsides of emotions, situations where emotions affect the agents utilities in an unpredicted good or bad way so that no or different equilibria are achieved.

The first possible outcome is the one with the most observable emotions: negotiation “fails”. The young professional in our example is annoyed by the way negotiation goes and eventually refuses to take the job. She leaves and neither she nor the firm is very happy about that.

The second outcome is that the negotiation had led to an agreement, but an agreement that leaves one or both players with positive or negative emotions.

One scenario in this case is a rather quiet one. She accepts a relatively low wage (or her boss accepts the high wage) and then one of them will constantly not be completely happy at the situation. If that does not have any effect on the other player’s emotion in a way that both agree on a new agreement eventually, an unhappy player will leave the firm sooner or later.

A more interesting scenario is one where the emotions of one player do affect the other party. For example, the sad end of the story would be if the young adult agreed on a relatively low wage, but would show her angry attitude at work ever after. The firm, the colleagues and as their representative the boss would suffer from that event. In theory, this situation would look the same the other way around, if the boss agreed on a high wage and would then show his anger, which would then badly contrast with the joy of the employee earning high wages.

A) Bargaining theory

This chapter provides a basic overview of the theoretical background of bargaining in the field of game theory.

1. An introduction into game theory

Game theory was introduced by the idea that conflict could be mathematically analyzed and it is still a very young science. (Rasmusen (2013), p.1). It became known to a bigger audience when politicians found out that that the tools of game theory could help to predict the behavior of companies or other countries that they are in relationship with (Ritzberger (2002), pp. 4, 5). In contrast to the neo-classical economic theory, in games, participants called players are aware of the impact of their decisions. Each game has an outcome that is derived by the strategies players use when trying to get the preferred outcome (Ritzberger (2002), p. 8).

i. A simple game and dominant strategies

Let us assume that two rational players play a simple game. Both players choose their strategies first and then reveal the chosen strategy. Both strategies combined lead to one payoff out of the possible multiple outcomes of the game. Possible payoffs of a simple game are given in the chart below.

Table 1: a simple game

illustration not visible in this excerpt

Player A may choose top or bottom and will receive the payoff of “1”, “0”, “1” or “2” (clockwise, starting in the intersection of “Left” and “Top”). If she acts rationally, she will choose “Bottom” as the expected payoff is “1.5” and will always be equal to or higher than the expected payoff from choosing “Top”. Player B’s expected payoff will be highest if he chooses “Left”, because then he will receive either “2” or “1”. The combined strategies will lead to “Bottom” and “Left”. Both players will always choose those dominant strategies. In order to be defined as “dominant”, a strategy has to be the best choice for one player, independent of any option the other player could choose. Mathematically spoken, player i’s strategy Abbildung in dieser Leseprobe nicht enthaltenis dominant if its payoff Abbildung in dieser Leseprobe nicht enthaltenis higher than the payoff of any other strategyAbbildung in dieser Leseprobe nicht enthalten. If there is a dominant group of strategies, those will always be the equilibrium-solution to the game. To derive such a solution, one iteratively eliminates non-dominant strategies. (Varian (2004), p.510, 511; Gibbons (1992), p. 3, 4)

Player B will get the payoff of “1” if choosing the dominant strategy. Player A knows that Player B wants to maximize his payoff and thinks of possible ways for B to do so, besides the dominant strategy. Player B could threat A to go “Right” if A chooses “Top”. But Player A will nevertheless pick “Bottom”: even in case B carries out the threat, she will receive the payoff of “1” while B’s payoff will shrink to “0”. As a rational player, B always prefers “1” to “0” and will therefore never threat A, as his threat is not credible.

ii. The Nash equilibrium and the sub-game perfect equilibrium

Dominant strategies do not exist for all possible games. If the simple game mentioned above were changed in the way that the payoff of “Right” and “Top” would be “0” for both players and the same would be the case for the interconnection “Bottom” and “Left”, there would not be one strategy leading to the highest payoff. Both players would have to coordinate their choice to meet at “Top” and “Left” or “Bottom” and “right”. This means both players would choose a strategy depending on the action of the other one. This can by no means be called dominant. But having once decided on a strategy that will give them both payoffs in this case, they will both not have any incentive to deviate from their strategy. Not even the player who will get the smaller payoff, as rationality is still assumed to be fundamental and envy etc. does not play a role (yet). To sum up: if player A’s decision is optimal, given players B’s choice and vice versa, they both will not deviate, and the game will end in equilibrium and this equilibrium is called Nash-equilibrium, named after the mathematician who got the Nobel Prize for this discovery. Generally speaking: if two players (A, B) play a game and A’s possible choices are while B’s choices are , there is a best answer for every and vice versa a best answer for every . A Nash-equilibrium is a pair of best strategies so that:

(Varian (2004), p.511, 512, 526). The Nash-equilibria in the modified example here are: “Top”, “Left” and “Bottom”, “Right”.

Let us modify the example one more time:

Table 2: the simple market game

illustration not visible in this excerpt

In this example, we think of a market that is dominated by one monopoly, player A. She earns “4” if she produces a monopoly-output, but that would hurt her in case she actually had a competitor, and than the monopoly-output would give her “-3”. So the top-row in this game is player A’s output in case of her monopoly. The bottom row will be her payoff, if she produces the smaller competitive output. Now the entrepreneur B thinks about entering the market. If he does so, he will share the market’s value of “2” with player A in case player A reduces her production. If player A tried to fight off player B, both would lose money. Player A wants to pick “Monopoly/Fight” if B chooses “Stay Out” and “Accommodate” if B picks “Enter”. Player B prefers “Stay Out” as an answer to “Monopoly/Fight” and “Enter” when A picks “Accommodate”. So we have two Nash-equilibria: “Monopoly/Fight”-“Stay Out” and “Accommodate”-“Enter”. If the game is carried out sequentially, and in this example this seems natural, one of those two equilibria will not be realistic. To rule out incredible threats, the Nash-equilibrium is extended to a concept called “Sub-game-perfect”-equilibrium. Such a threat would be A’s announcement to pick “Monopoly”, no matter what will happen. Such a game would look like Figure 1: if B chose to enter the market, A could defend her position and face an even bigger loss than B, while tolerating a new competitor what would leave both with a positive payoff. Therefore, it is unrealistic that A will choose the no-matter-what strategy and B will enter the market anyway, leaving only one equilibrium: B picks “Enter”, A picks “Accommodate”. This is called sub-game-perfect, because every sub-game is solved by a Nash-equilibrium and therefore the whole game is solved with a Nash-equilibrium. The game has two sub-games: the one where B decides and the one where A decides (the second nodes in figure one).

illustration not visible in this excerpt

Figure 1: sequential game with two sub-games

B’s assumption on the basis of backward induction that A will act when defending the market solves the first game and thus A’s rationality solves the second game. A’s solution to turn that game could be a contract, where A binds himself to play “Monopoly/Fight” and therefore B will never enter the market. In this case, the Nash-Equilibrium of “Enter”, “Accommodate” would be ruled out. (Gravelle, Rees (2004), p. 365-367; Varian (2004), p.519-521)

iii. Pure and mixed strategies

Sometimes, there are no pure strategies that lead to an equilibrium. Let’s look at the simple game again, modified for the last time:

Table 3: modified simple game for mixed strategies

illustration not visible in this excerpt

If player A chooses “Top”, player B wants to answer with “Right”. But then player A will prefer do deviate to “Bottom”, which will make Player B want to deviate to “Left”. From there, A wants to deviate to “Top” again. The solution to this will be to allow both players to choose the probability with which they are allowed to pick pure strategies. Let us assume, player A plays “Top” with probability x, while player B plays “Right” with probability y. The expected payoffs will then be:

A mixed strategy Nash-equilibrium is a pair of mixed strategies so that for all and for all y .

Now what is A’s best x for a given y? The partial derivative of with respect to x is:

Depending on y being smaller, equal or bigger than , x should be set to 0 for , 1 for and A should be indifferent for . (cf. Gravelle, Rees (2004), p. 385-387; Varian (2004), p. 527).

2. Games in bargaining theory

Bargaining is a matter that we face every day. As bargaining is often not reasonable in the sense that the typical assumption of rationality, that economic theory uses as basis, can be applied, it was hard to be described. That has changed with the applications of game theory. Suddenly, encounters between players that act strategically to force their preferred outcome of a bargaining situation, driven by complex motives, can be described. (cf. Dixit, Skeath (1999), p. 522)

Bargaining can be seen as a model for price determination, but not as known from micro-economics where there are many players involved on the demand side as well on the supply side. In markets with few players on both sides, player would start to bargain (Rasmusen, (2013), p. 357).

Game theory is divided into two branches, cooperative game theory and non-cooperative game theory (Ritzenberger (2002), pp. 7, 8). Both branches are addressed in this chapter, but the focus is set on non-cooperative games as those are much more useful when trying to include emotions.

i. Cooperative bargaining and the Nash bargaining solution

Let us assume a bargaining problem with two players and a set of possible agreements P and a disagreement point d , which will be the outcome of the game if the parties involved fail to achieve an agreement. An agreement in P produces a utility payoff, and the set of utilities is U . The bargainers’ preferences are given by a utility cardinal function for each player. This bargaining problem is called a cooperative bargaining game if three assumption are satisfied:

- U is closed, convex and bounded.
- The players can agree to disagree (so the utility of d is available in the set U ).
- The game is interesting: both players can be better off when agreeing and therefore there is an incentive to bargain.

A bargaining solution is a rule that can be applied to any kind of cooperative bargaining problem and picks a unique point in U . Nash argues that if such a problem satisfies four assumptions, there is only one rule which is called Nash bargaining solution.

- The solution chosen by the rule must be efficient. No other solution must leave one party better off and the other party at least not worse off.
- The game has to be linearly invariant. If the bargaining game is relabeled by a linear transformation of the players’ utility function, then transforming the solution in the same way must be the only solution to the transformed game.
- The game must be symmetric. That means that the players have the same preferences and receive the same utility. Due to linear invariance, this is easy to achieve by transforming one or both utility functions and the solution will not be essentially affected.
- No irrelevant alternatives. This is quite intuitive if one imagines bargaining problems as two parties ruling out agreements step by step until only the best solution is left. Bargaining power is not considered in this theory so the outcome does not depend on one player’s worst or best possible solution. Ruling those out can’t change the outcome here.

Let now and be the utility of the only two players A and B involved in a cooperative bargaining game and and are both players’ utilities when no agreement has been made (point d ). The Nash-solution is now the solution to the following maximization problem of a Nash-product:

illustration not visible in this excerpt

so that , and . (Gravelle, Rees (2004), pp. 387-395).

Although it would be appealing, this is not the only way to make assumptions about games and derive unique solutions. Moreover, it leaves the process of bargaining unexamined. This theorem of the branch of cooperative game theory implies that the outcome leads to pareto-efficiency. But parties in real bargaining situations will be in different circumstances and some players are likely to have greater power than others. And those more powerful players use their strength to maximize their own shares in competitive environments. They follow strategies which will not always lead to an pareto-efficient outcome, in order to be better off. This model does not consider strategies and it is therefore better advice to look at non-cooperative bargaining games if one wants to model real-life bargaining situations. (Rritzberger (2002) p. 8, Gravelle, Rees (2004), p. 395)

ii. The Rubinstein model: a non-cooperative game

a. Introduction and example

The game which was introduced by Ariel Rubinstein in 1982 was so convincing and elegant that it created a big impact in game theory (Varian (2004), p. 550) and ha been the standard model of non cooperative bargaining ever since. The model itself is rather easy: two players have to share a certain valuable token desired by both of them, usually referred to as a pie or as one dollar. Both player have to give offers of shares in an alternating order, and both can accept an offer and accept the shares proposed by the other player (the game ends right here) or reject an offer and make one of their own (Levin (2002), p.2). Several restrictions can be made, such as that the game has to end after three turns and that both players do not receive anything if they do not agree on a solution how to share the pie (Varian (2004), p.548). The assumptions Rubinstein (1982) made for the family of models he presented were the following:

A1: “`pie´ is valuable”
A2: “time is valuable”
A3: “continuity”
A4: “stationarity” (whether I prefer share X of the dollar now or share Y in `one period after now´ does not depend on the period)
A5: “the larger the portion the more compensation a player needs for a delay of one period to be immaterial to him”

To meet these assumptions, a utility function for each player is implied which here is simply the payoff of the bargain solution and therefore the received share of the underlying value. Moreover, each player has a discount factor which is α for player A and β for player B, so that 0 ≤ (α, β) ≤ 1. For example, player A’s utility in time period 0 of a share x received in t=2 would be = .

Figure 2 shows the game in its extensive form: player A, purple as before, starts with the first offer, her first node is purple. Player B can now decide to either accept the proposed shares in turn t=0, consumption and therefore payoff is resolved immediatley. If he rejects and makes a counteroffer in turn t=1, player A can decide to either propose again in the next period t=2, or accept and consume in t=1. The payoff is discounted to represent the player’s impatience, utility in the future will have less value from the present perspective.

Figure 2: Rubinstein-game in extensive form

If we stick to the example given above, in which two players share one dollar and do not get anything if they do not agree, with the game ending after three periods, we can solve this game by iterating from the end. Both players know the discount-factor of the other player (and know about him/her knowing). Player A, who starts with the first offer, also has the last turn in this game. If the game has not come to an end yet, she can make any offer and a rational player B has to accept, as long as his share is slightly (but not measurably) higher than zero. Player B anticipates this “do as you like-offer” in t=1 and knows that the value of player A’s share in the current period is α dollars. So he proposes α Dollar to his counterpart, receiving 1-α Dollar. That again is foreseen by player A in t=0 who makes an offer of β (1-α) to player B, knowing that he will accept. Her outcome is therefore (1-β (1-α)). (Varian (2004), pp. 248,249)

[...]