Friday, October 9, 2015

Preferences, Property Rights, and Anonymity in Bargaining Games - Ultimatum Games

By: Elizabeth Hoffman, Kevin McCabe, Keith Shachat, and Vernon Smith


Non-cooperative, non-repeated game theory is about strangers with no shared history meeting and interacting strategically in their individual self-interests according to well specified rules and payoffs, and then never meet again.  Experimental studies of these two-person bargaining experiments are generally not consistent with the game theoretic predictions, and they do not always replicate across subject populations, particularly in the absence of monetary rewards. 

In recent experimental research on ultimatum games has found that first movers tend to offer more to their counterparts than non-cooperative game theory would predict.  The common offer is half the surplus to be divided, although non-cooperative game theory would suggest an offer by the first mover of the minimum positive amount that is feasible.   

In an ultimatum game an amount of money M is to be divided between 2 subjects.  One, the designated proposer, announces a split of M – X to the proposer and X to the proposer’s counterpart.  After the proposal is made, the counterpart either accepts or rejects it.  If the counterpart accepts, then the proposal is carried out, but if it is rejected then both the proposer and the counterpart get zero.  A rational and non-overly materialistic counterpart should accept the offer X = e > 0, where e is the minimum unit of account.  The equilibrium prediction is for the proposer to offer X = e and for the counterpart to accept. 

Experiments should that first mover proposers in these bargaining games offer more to their counterparts than non-cooperative game theory leads one to expect.  The tendency toward an equal split is often described as “fairness” or “social norms” of distributive justice, but they do not explain the phenomenon in terms of testable fundamentals. 

A second game adjusted the parameters to create a “posted offer” scenario.  The seller begins the process by choosing a price; this price is communicated to the buyer, who then chooses the quantity, thus ending the game.  Consequently, the seller makes an ultimatum price offer to the buyer.  This differed from the first experiments in that:
  1. All bargaining was described as a buyer/seller transaction
  2. The equilibrium yielded more than an e payoff to the buyer
  3. Both sellers and buyers had multiple price/quantity choices available, but the buyer was free to reject the price offer by choosing a zero quantity.

In the ultimatum game, the proposer must form expectations about their counterpart’s reservation value.  Thus, a risk averse proposer may give his or her counterpart more than is predicted by non-cooperative theory in order to insure acceptance of the proposal.
The article notes that randomization of assigned types may not be neutral as subjects can interpret their assignment as the experimenter treating them fairly, thus facilitators may induce a “fair response,” feeling that they should be fair since the facilitator was fair.  If the first mover earns their right to their role, offers are smaller.  When this earned entitlement is combined with exchange, less than 45% of the first movers offer $4 or more out of $10.  Random Entitlement equated to over 85% offering $4 out of $10.  The strategic/expectational character of ultimatum games makes it impossible to conclude from offer data alone whether offers in excess of $1 are due to other regarding preferences or to the first mover’s concern that their offer might be rejected unless it is deemed satisfactory by the second mover.   


The experiment followed instructional procedures for inter-subject anonymity as a partial control for the effect of social influences on choice. The ultimatum game in game theory does not require a knowledgeable experimenter and facilitators must be aware of pregame treatments and careful instruction.  It was interesting to note that the ultimatum experiments, randomization of assigned types may not be neutral and could induce a “fair response.”  This implies that first mover offers are sensitive to the instructional setting of the experiment.  The results suggest that behaviors deemed as “fairness” are actually a social concern for what others may think and for being held in high regard by others.   The article interprets offers in ultimatum games as appearing to be determined by strategic and expectations considerations rather than the result of an autonomous private preference for equity.



  1. Do you think this study would have the same results if it was replicated today (the original study was published in 1994) or in a different country? Or will the instructional setting play a role on the results regardless of culture? I remember one of the discussions in class was how we like people who are more like ourselves, and it makes me question whether our inherent belief systems interfere with our actions when under observation. Though the study attempted to control for this aspect with anonymity, the environment likely played a role in the results.

    1. I definitely agree with your observation. I think that the study would likely have similar results if conducted today since the it has more to do with human behaviors which would not vary over a few decades. Though I do see where results could vary based on the country or type of culture the study is conducted in. The study did control for anonymity to a degree, but I could see environmental factors contributing to results as well. I did not consider this perspective so I am glad you pointed it out.

  2. I think this is a great example of how our social norms and constructs can greatly undermine classical game theory. The idea of purely rational actors is a helpful one in trying to conceptualize a situation, but when real analysis is being conducted there are many more human variables to confuse what appeared cut and dry on paper.

  3. I agree. I think the study could vary depending on cultures especially when comparing individuals who never bargain to those who do. I also really liked how you summed up the math in this example making it easy to understand and visualize for those of us who are numerically challenged.