How Time Preferences and Level of Risk Can Affect the Nash Bargaining Solution
The Nash Bargaining Solution (NBS), as presented in lecture, was a static model in which both the utility and bargaining power of each person was fixed in time. In reality, however, these variables are constantly changing as each bargainer reevaluates their position. Two variables that contribute to this time-dependence are the bargainers’ time preference and level of risk. The article referenced below constructs a model for evaluating how these two variables can offset the equilibrium solution – leading to a different result than that of the static NSB.
In this model, time preferences are defined as the opportunity cost one is willing to sustain while the dispute is being settled. Similarly, level of risk is defined as the probability that negotiations breakdown as the dispute carries on. This paper argues that the utility functions which incorporate these two variables have the property of downward concavity. This is interpreted as the longer it takes to reach a deal, the more opportunity cost each person must sustain for less utility (Figure 1). As time progresses, it also becomes more likely for negotiations between the two parties to breakdown; this in turn reduces the rate of utility as the risk of not reaching a deal rises over longer periods. One example is the longer a dispute progress, the more time a third-party would have to reach a deal with one of the two bargainers.
In the short-term (defined as the length of the bargaining period tending to zero), the paper shows that the time-dependent variables above have little effect on the bargaining solution, which results in the model computing the static NBS. However, as the bargaining period increases, the model predicts that the equilibrium solution becomes offset in favor of one of the bargainers. This has to do with each bargainers’ utility function and the individual rates of utility increase as a function of time. The player with the smaller rate of utility increase will lose bargaining power at a faster rate, leading them to concede prior to the other bargainer (since opportunity cost is also continuously accumulating). The paper states that the equilibrium solution, which arises from this time-dependent model, is different from that of the static NBS for long bargaining time periods.
Although it requires the derivation of each bargainer’s utility function to determine who will yield first, there are general characteristics within the model that affect the likelihood of yielding and the rate of decrease in bargaining power. The first has to do with who makes the initial proposal. The model predicts that the bargainer who makes the first move is less likely to yield than the opposing bargainer. This can be interpreted as the proposing bargainer being less happy with the status quo, and thus, being more accepting of the opportunity cost associated with the dispute. Similar reasoning suggests that the longer it takes the responding bargainer to counteroffer, the less powerful they become as opportunity costs begin to accumulate. Finally, the model predicts that the greater a player perceives negotiation breakdown, the lower their bargaining power becomes; this has to do with one player being more desperate to make a deal than the other. It should be noted that the above discussion is only valid when the value of each player’s outside option is below the value of engaging in a deal.
Figure 1: The above image demonstrates how the rate of increase in utility decreases over time, leading to a loss in bargaining power as the time without a deal increases.
Source:
https://www.jstor.org/stable/pdf/2555382.pdfrefreqid=excelsior:31f729663373f622b3f64e15c135d508