Cooperative game theory assumes that groups of players, called coalitions, are the primary units of decision-making, and may enforce cooperative behavior. Consequently, cooperative games can be seen as a competition between coalitions of players, rather than between individual players. The basic assumption in cooperative game theory is that the grand coalition, that is the group consisting of all players, will form. One of the main research questions in cooperative game theory is how to allocate in some fair way the payoff of the grand coalition among the players. The answer to this question is related to a solution concept which, roughly speaking, is a vector that represents the allocation to each player. Different solution concepts based on different notions of fairness have been proposed in the cooperative game theory literature.
ย An example of a cooperative game is a joint venture of several companies who band together to form a group (collusioin).ย
CORE
In cooperative game theory, the core is the set of feasible allocations or imputations where no coalition of agents can benefit by breaking away from the grand coalition. One can think of the core corresponding to situations where it is possible to sustain cooperation among all agents.ย
Consider a group of n miners, who have discovered large bars of gold. If two miners can carry one piece of gold, then the payoff of a coalition S is
๐ฃ(๐)={|๐|/2,if |๐| is even;(|๐|โ1)/2,if |๐| is odd.
If there are more than two miners and there is an even number of miners, then the core consists of the single payoff where each miner gets 1/2. If there is an odd number of miners, then the core is empty.
For the moment ignore shoe sizes: a pair consists of a left and a right shoe, which can then be sold for โฌ10. Consider a game with 2001 players: 1000 of them have 1 left shoe, 1001 have 1 right shoe. The core of this game is somewhat surprising: it consists of a single imputation that gives 10 to those having a (scarce) left shoe, and 0 to those owning an (oversupplied) right shoe. No coalition can block this outcome, because no left shoe owner will accept less than 10, and any imputation that pays a positive amount to any right shoe owner must pay less than 10000 in total to the other players, who can get 10000 on their own. So, there is just one imputation in the core.
The message remains the same, even if we increase the numbers as long as left shoes are scarcer. The core has been criticized for being so extremely sensitive to oversupply of one type of player.
The Shapley value is a solution concept used in game theory that involves fairly distributing both gains and costs to several actors working in a coalition. Game theory is when two or more players or factors are involved in a strategy to achieve a desired outcome or payoff.
A famous example of the Shapley value in practice is the airport problem. In the problem, an airport needs to be built in order to accommodate a range of aircraft that require different lengths of runway. The question is how to distribute the costs of the airport to all actors in an equitable manner.
The solution is simply to spread the marginal cost of each required length of runway among all the actors needing a runway at least that long. In the end, actors requiring a shorter runway pay less, and those needing a longer runway pay more; however, none of the actors pay as much as they would have if they had chosen not to cooperate.
Shapley values help with marketing analytics. A company selling its product on its website will likely have different touchpoints, which are ways for customers to engage with the company and drive them to ultimately buy their product.
For example, a company might have various marketing channels to attract potential customers, such as social media, paid advertising, and email marketing campaigns. The Shapley value can be applied here, assigning each marketing channel as "players," with the "payoff" being the purchase of the product. By assigning values to each channel, Shapley value analysis can help determine what channels get the credit for the online purchase.
In theory, a player can be a product sold in a store, an item on a restaurant menu, a party injured in an auto accident, or a group of investors in a lottery ticket fund. The Shapley value can be applied in economic models, product line distributions, procurement measures for embassies and industries, market mix models, and calculations for tort damages. Strategists are continuously discovering new methods to use the solution.