Altruistic Preferences in the Alternating Offer Bargaining Game

Altruistic Preferences in the Alternating Offer Bargaining Game Brian Armstrong

Overview • Experimental results from the Ultimatum Game indicate that people do not possess classical rational preferences • This has led to the development of inequity averse preference models used to interpret the Ultimatum Game • However, the Ultimatum Game is a unique bargaining framework in that it does not allow repeated interactions • The goal of my study is to examine the theoretical implications of applying inequity averse preferences to the Rubinstein Alternating Offer Bargaining framework • I will conclude with the following results • A given player’s equilibrium allocation is increasing in their level of “spite” and decreasing in their level of “generosity” • The importance of altruistic inequity averse preferences is not monotonic in the discount rate

Outline 1. Ultimatum Game 2. Altruistic Preferences 3. Extension to Infinite Horizon Repeated Bargaining 4. Results 5. Conclusion

Ultimatum Game Framework of Ultimatum Game • Two players (Proposer and Receiver) are bargaining over how to partition a “pie” of size 1 • Proposer makes an initial proposal of xε [0, 1] • Receiver may choose to either accept or reject the initial proposal • Accept: Proposer receives x, Receiver receives 1-x • Reject: Proposer receives 0, Receiver receives 0 • Nash Equilibrium: x* = 1, Receiver Accepts • Proposer receives 1 (entire “pie”), Receiver receives 0 • Solution based on Receiver’s indifference between rejecting and receiving 0 or accepting allocation of 0

Ultimatum Game Experimental Results Guth et. al., 1982: An Experimental Analysis of Ultimatum Bargaining Clearly, the results do not reflect the Nash Equilibrium

Altruistic Preferences Fehr-Schmidt Model of Inequity Aversion • Xirepresents the proportion received by player i • Xjrepresents the proportion received by player j • If Xi and Xj are not equivalent, both players feel some form of disutility • The α parameter represents a player’s aversion to disadvantageous inequity • The β parameter represents a player’s aversion to advantageous inequity

Altruistic Preferences Inequity Aversion • α-value: Players choose actions out of spite when they feel they have been treated unfairly • Responders reject non-zero allocations in favor of zero allocation • β-value: Players choose actions that are socially acceptable – Players are concerned with the collective utility • Proposers offer generous non-zero allocations • Overall, if the solution is different from the fairness reference point, all players experience some form of disutility

Altruistic Preferences Application to Ultimatum Game

Altruistic Preferences Application to Ultimatum Game Equilibrium Solution Solution is only dependent on βi and αj

Extension to Infinite Horizon Repeated Bargaining Questions What do altruistic preferences predict in repeated bargaining? And more specifically: How do the altruistic preference parameters affect the equilibrium solution? How does the discount rate modulate the effects of fairness preferences on the equilibrium solution?

Extension to Infinite Horizon Repeated Bargaining Framework of Alternating Offer Bargaining Game • One player begins the bargaining by proposing a partition. The opposing player may either reject or accept the proposal. • If the proposal is accepted, the game ends there. • If the proposal is rejected, the game progresses to the next period and the original responder must now propose a partition. • As the bargaining proceeds to the next period, each player experiences a time cost of bargaining which is reflected through the fixed discounting factor δ. • This process continues until an agreement is reached.

Extension to Infinite Horizon Repeated Bargaining Method of Proof • Consider a time t where Player I proposes x=(xi, xj) • We assume that if Player J rejects x, then in time t+1, Player J will propose y=(yi, yj) • Similar to Rubinstein, our SPE will be constructed under the assumptions: • Each player is indifferent between acceptance and rejection • Every proposal is independent of time and history • Every proposal is immediately accepted in equilibrium

Extension to Infinite Horizon Repeated Bargaining Method of Proof T Therefore, we must solve the following system of inequalities-

Extension to Infinite Horizon Repeated Bargaining Method of Proof • But remember: We will immediately disregard cases (4) – (6)

Results Graph: Fixed Proposer

Results Symmetric Solution Intuition • Case (1) βi < ½, βj < ½ • As each player’s α value increases, their equilibrium allocation increases. A larger α value represents a more credible threat of rejection for a given offer. • As each player’s β value increases, their equilibrium allocation diminishes. The larger β value can be exploited by their opponent. • Case (3) βi < ½ , βj > ½ • The receiver’s β value is greater than ½, thus his optimal allocation is ½. Knowing this, the proposer understands that she can never receive less than ½, and therefore the solution is dependent solely on the proposer’s β value and the receiver’s α value.

Results Graph: Fixed Receiver

Results Symmetric Solution Intuition Case (1) βi < ½, βj < ½ As each player’s α value increases, their equilibrium allocation increases. A larger α value represents a more credible threat of rejection for a given offer. As each player’s β value increases, their equilibrium allocation diminishes. The larger β value can be exploited by their opponent Case (3) βi > ½ , βj < ½ The proposer’s optimal allocation is ½. Thus, the proposer will always propose an allocation of ½.

Results General Solution: Case (1) • Each player is allowed to have their own unique discount rate δ • The general solution has 3 possible Proposal Scenarios: • (a) Similar Proposer and Receiver: xi > xj, yj > yi • Players have similar altruistic parameters and/or discount rates • (b) Strong Proposer and Weak Receiver: xi > xj, yj< yi • Strong Proposer: High δ and low β • Weak Receiver: Low δ and low α • Given δ’s, the solution is dependent on the Proposer’s β value and the Receiver’s α value • (c) Weak Proposer and Strong Receiver: xi< xj, yj > yi • Weak Proposer: Low δ and low α • Strong Receiver: High δ and low β • Given δ’s, the solution is dependent on the Proposer’s α value and the Receiver’s β value

Results Proposal Scenario (b)

Results Proposal Scenario (c)

Results Question 1 How do the altruistic preference parameters affect the equilibrium solution? α Parameter - β Parameter - δ Parameter - If Symmetric - Clearly, the solution is dependent on much more than the Proposer’s β value and the Receiver’s α value

Results Question 2 How does the discount rate δ modulate the effects of fairness preferences on the equilibrium solution? The Ultimatum Game can be considered the limit of the Rubinstein Bargaining Game when the discount rate equals zero As we allow the discount rate to deviate from zero, we extend the horizon of our bargaining interaction By comparing the Rubinstein rational equilibrium solution to the symmetric altruistic equilibrium solution, we can investigate how the discount rate (and thus the horizon of the bargaining interaction) affects the deviation of the altruistic equilibrium from the rational equilibrium

Results Typical Altruistic Equilibrium Proposal

Results Abnormal Altruistic Equilibrium Proposal

Results Question 2 • Equilibrium proposal diminishes as δ increases • As δ goes to 1, both the altruistic and rational solutions approach ½ • Under typical altruistic parameters, the altruistic equilibrium proposal is less than the rational equilibrium proposal • As δ increases, the difference between the rational equilibrium and altruistic equilibrium decreases • Under abnormal altruistic parameters, the altruistic equilibrium proposal is actually greater than the rational equilibrium proposal • Difference is significant when δ is such that a counter-offer is feasible but not necessarily optimal

Conclusion Concluding Remarks • By extending to the alternating offer bargaining game, the SPE reflects a more complete version of altruism and fairness • Spiteful players can use their credible threat of rejection to secure a larger allocation • Players with concern for the overall good are exploited by their opponents and their allocation is diminished • As the discount rate approaches one, the effects of altruistic parameters decrease and the altruistic solution approaches the rational solution • Typically, the altruistic equilibrium proposal is below the rational equilibrium proposal. • However, under certain circumstances, the altruistic equilibrium proposal is above the rational equilibrium proposal as the altruistic player is able to leverage the credible threat of rejection

Conclusion Possible Extensions • Allow the fairness reference point to vary • Compare fixed time cost of bargaining vs discount rate • Introduce potential for breakdown in negotiations • Run an experiment to test the predictive power of the equilibrium

Additional Slides

Results Symmetric Solution

Results General Solution: Case (1)

Results General Solution: Case (1) Boundaries Proposal Scenario (b) Proposal Scenario (c)

Results Symmetric Altruistic Case (1) Solution Symmetric Rational Players Solution Difference

Section Title • Bullets

Altruistic Preferences in the Alternating Offer Bargaining Game