Nash Bargaining in Relationships
In the July 2017 Time article, “This One Simple Tool Could Save Your Relationship,” Chris Wilson proposes an application of Nash Bargaining to solve a common relationship problem: deciding what to spend money on. The game Wilson describes is a couple deciding to indulge in a spending splurge, but the two partners not agreeing on what to spend the money. He proposes a solution through Nash Bargaining where each partner rates how satisfied they will be (1-10) if all money is spent on their desired outcome, and rates how satisfied they will be (1-10) if no solution is reached and no money is spent. Then, to decide how to spend the money, Wilson describes this procedure (edited for brevity):
The output of Nash’s formula is the amount of money each person gets, adding up to the total. To measure the quality of the bargain, it takes the percentage of the total that each person got and multiplies it by the degree of happiness they would have experienced if they got all of it. So if a person rates their ideal satisfaction as an 8 on the 1-to-10 scale, and gets 75% of the money, their happiness is considered to be 6. It then subtracts the baseline happiness they would feel if negotiations were to fall apart to measure how much more satisfied they are with the agreement than they would be if there was none. Last, it takes this figure for both parties and multiplies them together, a value called the “Nash Product.” The magic of Nash’s equation is that it can determine the split that would produce the highest value for this product.
This scenario describes a different type of Nash Bargaining compared to what we discussed in class. Whereas in class we defined Nash Bargaining as two players who bargain by evenly splitting the surplus generated by their deal (and ultimately receiving half the surplus in addition to the value of their outside option), Wilson defines this bargain as maximizing a product. To define this product, we can let HA = player A’s happiness if they got all the money, HB = player B’s happiness in they got all the money, NA = A’s happiness if no deal were reached, NB = B’s happiness if no deal were reached (all being fractions ranging from 0-1). Additionally, let a = the value A will receive if a deal is reached, and 1-a = the value B will receive if a deal s reached (letting the total value to be divided up be equal to 1). Using this notation, the Nash Bargaining Wilson describes maximizes the product of:
(a*HA – NA) * ((1-a)*HB – NB), or, simplified:
(a*HA – NA) * (HB – NB – a*HB).
This product does not have exactly the same properties as the Nash Bargaining we discussed in class, but it does have some of the same principals. For example, we learned how having a higher outside option gives an individual more power in a bargaining game. Here, that principal still applies: having a higher happiness rating of reaching no deal (N) leads to a player receiving a higher proportion of the value to be split, because the player can use their ambivalence to leverage their partner into making a deal in their favor. However, here the math is more complicated to figure out what value each player will receive from the deal.
Link here! Check out the interactive tool! Maybe even use it to solve problems in your relationship.
— Jack Schluger, Cornell ’21