How will the NBA offseason play out? The most interesting storyline concerns the Miami Heat. In 2010, LeBron James, Chris Bosh, and Dwyane Wade all took less money and coordinated to play for the Miami Heat. The “superteam” reached the NBA Finals each year and won 2 championships. This year, facing a lopsided loss to the San Antonio Spurs, the three stars have all opted out of their contracts. Each player faces a conflict between a personal interest to get more money versus a joint interest of taking less money, allowing the team flexibility to sign other players. The situation makes for a great example of the game theory topic of fair division.

As I’ve written about before, fair division is in the eye of the beholder. Take the simple situation of splitting the bill. Some groups split the bill evenly, others ask each to pay for what was ordered, and some have other conventions–like alternating who picks up the tab. Even more unusual divisions are possible, as exemplified in a bankruptcy problem in the Talmud.

What LeBron James, Dwyane Wade, and Chris Bosh feel are fair is personal, and that will affect the final negotiation. However, as a starting point, it is useful to consider an objective standard. In this post, we answer the following question: if the Big 3 sit down and agree to divide up available money according to their value to the team’s playoff success, how much should each player get? The game theory concept of the Shapley value provides a unique answer with LeBron James getting most of the money, as explained after the break.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Cooperative Games

Famous games like the prisoner’s dilemma, the game of chicken, and the battle of the sexes are examples of non-cooperative game theory. Each player is assumed to pursue selfish interests, and the players cannot make binding agreements. For example, the two prisoner’s might each say they will stay quiet, but this agreement cannot be enforced. In these games, the appropriate solution concept is the Nash equilibrium.

Contract negotiations and fair division problems are examples of a different subject in game theory, that of cooperative games. Players can either go at it alone or they can form coalitions with binding contracts. These games require a different equilibrium analysis. One solution concept is the Shapley value which loosely speaking gives a player value relative to the contribution to the group. Because the Big 3 have coordinated their actions in the past, it seems appropriate to model this as a cooperative game.

The Game

ASSUMING THEY ALL RE-SIGN WITH THE MIAMI HEAT…

Imagine the Big 3 have the following discussion. They want to play together, and they will agree to be paid in proportion to their value to the team. During negotiation, if the deal sounds unreasonable, each player can break off and go at it alone, or pairs of players might join forces and enter into an agreement.

The strength of each player’s position depends on his contribution to the team. We will have to make some assumptions about how much impact a player can have a team by himself, in pairs, and with all three together. Based on historical performance, here is how far we think each set of players can take a team next year.

{LeBron James} – Eastern Conference Finals

{Chris Bosh} – Second Round Playoffs

{Dwyane Wade} – First Round Playoffs

{LeBron James, Chris Bosh} – Eastern Conference Finals

{LeBron James, Dwyane Wade} – Eastern Conference Finals

{Chris Bosh, Dwyane Wade} – Second Round Playoffs

{LeBron James, Chris Bosh, Dwyane Wade} – NBA Finals

We can also represent the game in a diagram. Values next to a player are his individual worth, whereas values on lines joining two players are their joint worth. The value at the bottom of the figure–the intersection of all three lines–is the value to the group when all three join.

Here is how the possibilities can be read. LeBron James is a dominant player, and in a weakened Eastern Conference, he can probably power a team to the Conference Finals. Therefore, any set with him–even a set of him alone–is the Eastern Conference Finals.

Chris Bosh has been less of an impact player in recent years. But he is still a strong player, and we say that he could probably bring a team to the playoffs and win one round.

Dwyane Wade has been facing a lot of injuries, so this is the hardest one to model. For sake of argument, imagine he is healthy enough to take a team to the playoffs but due to the chance of injury he cannot be counted on for any more (not to underestimate him–this is just one possible scenario).

When all three are together, the group can perform even better and get to the NBA Finals.

The point of this model is that each player offers some worth to the team, and joining up together is better than going at it alone.

Principles For Fair Division

Fair division depends on social customs and rules. In 1953, Lloyd Shapley proposed a set of rules for fair division and then proved these axioms lead to a unique division. In part for this work, he shared the Nobel Prize in 2012. Here are the four rules and why they make sense.

1. Efficiency Axiom: The Entire Value To the Group Is Divided

This rule is about making sure the group achieves its highest possible value and does not waste any of that value. In the current game, if the players care most about getting to the NBA Finals, and they cannot do it any other way, they should find a way to stay on the same team.

2. Null Player Axiom: A Player That Adds No Value To Any Group Gets Nothing

This rule is intuitive that players who contribute no value should not be allocated a share of the prize. For example, let’s say you add me as a player to the game {Presh}. The value of {Presh} is zero since I cannot improve an NBA team’s performance on the court, and if you consider any set {Presh, other players} the value would be the same as {other players}. So I would be a null player and should not get any share of the value.

3. Symmetry Axiom: Players Who Add Exactly The Same Value Get the Same Shares

A left and a right shoe are equally valuable to footwear, so if they were priced separately instead of in pairs, this rule would say they should be priced at exactly the same cost. In the NBA, players that perform the same role on the team–in any arrangement of players–should be treated as having the same value.

4. Additivity Axiom: If Two Games “Add Up” To a Single Game, The Values In Each Game Add Up To The Single Game

Imagine you and friends go to lunch and then dinner and split the check. This rule roughly says that the amount paid for lunch and dinner–two games–should equal the amount paid for the entire day treated as one game of lunch and dinner. The reason is that you can obviously split the check for each meal or you can split it for the entire day. The division should be the same.

The Shapley Value

These four rules are reasonable. And the remarkable part is they uniquely determine a fair division, which is called the Shapley value.

To calculate the Shapley values, we need to place numerical values on each of the outcomes. Let’s say the main goal is to get to the NBA Finals (100), and each round of the playoffs is worth its percentage of completing to the goal. So the first round is 25 percent, the second round 50 percent, and the Eastern Conference Finals 75 percent. Here is the game with these values.

{LeBron James} – 75

{Chris Bosh} – 50

{Dwyane Wade} – 25

{LeBron James, Chris Bosh} – 75

{LeBron James, Dwyane Wade} – 75

{Chris Bosh, Dwyane Wade} – 50

{LeBron James, Chris Bosh, Dwyane Wade} – 100

How will this game play out? To start, let’s see how the axioms play a role in determining the outcome.

First, by the efficiency axiom, the players should be able to join together to reach the highest value of 100, and that entire value gets divided. Second, there are no null players or symmetric players in the game presented. However the game can be decomposed into smaller games which do have null and symmetric players, and those values are determined by the axioms. Finally, by the additivity axiom, the value of this game must equal the sum of the values of those smaller games.

Now, let’s get a numerical value.

Calculating the Shapley Value

The math of how to calculate the Shapley value is beyond this post. But the algorithm to calculate is not too difficult and we will present it here.

The Shapley value can be calculated as the average of the marginal contribution of each player to the group across all possible orderings.

With Miami’s Big 3, the team can sign any of the 3 players first, then any of the remaining 2 players, and finally there is just one player to sign. So there are 6 = 3(2)(1) possible ways the players can be signed. And in general for n players there are n! ways the players can be ordered.

Let’s calculate the Shapley value for LeBron James (J), Chris Bosh (B), and Dwyane Wade (W). What do we mean by marginal contribution? It is the amount an extra player can add to the team. For example, imagine the team signs James, and it is uncertain that it can sign Bosh or Wade. James’s marginal contribution to the team is 75, because by himself he can get 75. If the team then adds on Bosh, the pair of James and Bosh is still worth 75. So Bosh’s marginal contribution in this ordering is 0. If Wade gets on board, then the trio can get to the Finals which is worth 100. So Wade’s marginal contribution is 25 in this ordering.

The Shapley value is calculated by repeating this process for every possible ordering and then averaging out the value. Here is a table of the 6 orderings, with each player’s marginal contribution, and the average value.

The Shapley value for the game has LeBron James with a 54.17 percent share, Chris Bosh with a 29.17 percent share, and Dwyane Wade with a 16.67 percent share.

Translation Into Contracts

The Miami Heat are reported to have about $55 million of cap space, of which $10 might go to adding another player and the remaining $45 million going to the Big 3.

If the $45 million were divided according to the Shapley values, then LeBron James would get $24.4 million, Chris Bosh $13.1 million, and Dwyane Wade $7.5 million. The calculation is quite harsh to Wade who was on track to make $20 million this year, and even Bosh could get more money from other teams. Rumors are that Wade and James will take something more like $15 million for the current year. So the model is probably understating Bosh’s and Wade’s contribution to the team, and one could change the coalition values to find out other results.

If the Big 3 played game theory according to this model, then LeBron James would get $24.4 million. As it turns out, this number is about in line with LeBron’s possible max salary of $20.7 million with the Heat.

There are obviously many caveats to this model, as the NBA has many contract rules, other teams can offer contracts, and the players would be looking for multi-year deals which was completely ignored.

Nonetheless, it is amusing the numbers worked out so closely to what LeBron is rumored to be asking for. The Big 3 are not actually playing game theory to determine their contracts, but in a way, it would be interesting if they did.

Further Reading

Insights Into Game Theory by Ein-Ya Gura and Michael B. Maschler is a fantastic book that explains the game theory of matching, Arrow’s voting paradox, the Shapley value, and the bankruptcy problem in the Talmud. The text builds up on simple examples and exercises and suddenly you just understand everything.