Game Theory & Bargaining

High-level overview of bargaining problems in sports.


Today I will cover the topic of bargaining and how it applies to game theory. Throughout our lives we engage in instances of bargaining whether it be work related, familial, or social. In this blog, I will focus on bargaining related to the NFL.

Recently, the NFL collective bargaining agreement has been under intense scrutiny due to the powers of the NFL commissioner. Let’s assume the current collective bargaining agreement has expired and talks have gone nowhere. The lack of an agreement runs the risk of spilling into the NFL season resulting in billions lost in revenue as the dispute continues.

Both the NFL and NFLPA have agreed to ultimatum bargaining (take it or leave it). A judge has been appointed to specify the rules of the negotiation. The overall amount of the “pie” has been determined to be 2.25 billion dollars (4 times the amount is more realistic).

Here are the rules:

  1. The players will begin the negotiations by making an offer to the owners. The offer will state the proposed split of the pie.
  2. The owners can either accept or reject the offer. If the owners accept, the negotiation ends and the pie is split as specified by the players. If the owners reject the offer, the owners would then make an offer to the players on how to split the pie.
  3. Each time a rejection occurs, 250 million is lost in game revenues resulting in the entire pie shrinking.
  4. The players can either accept or reject the owners’ offer. If accepted, the negotiation ends. If rejected, another offer is made by the players. The process continues until either an offer is accepted or all the $2,250,000,000 is lost.

So to solve this bargaining game, we can use a combination of probability and backwards induction.


Above is the scenario of the game through five rounds. The players have first movers advantage due to the rules from the judge.

This is represented in the following probability formula: (n+1)/(2n).

“n” represents the number of rounds in the game, which is determined by the formula: Total Amount/Loss Revenue per reject or $2,250,000,000/$250,000,000 = 9.

So the optimal percentage share for the players would be (9+1)/(2*9) = 56% or .56

The owners have a disadvantage represented by the formula: (n-1)/(2n) = (9-1)/(2*9) = 44% or .44

So the optimal choice for the players is to present a 56/44 split of the pie. Now if we work backwards we can see that it is in the owners best interest to accept the initial offer. The reason is because once the owners reject the initial offer, the pie shrinks by 11% (-250 million for each offer rejected). In the 2nd round of negotiations the players will reject the 2nd offer due having more favorable terms in the 3rd round. Once we enter the 3rd round, it doesn’t get any better for the owners as the pie continues to shrink.

This is a simple representation of bargaining which is more complex in the real world. In addition, the NFL would have more power and influence in negotiations over the players as seen in recent collective bargaining agreements.

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