1. Consider a very simple setting: a fish stock is harvested by two symmetric (identical) players, which can be fishermen or fleets. 2.1 A Simple Non-cooperative Game This game is based on the classical Gordon-Schaefer bioeconomic model. Fish stock dynamics:

2. What is the expression of the steady-state relation between the stock and the fishing effort ? The economic dimension of the fishery is represented by the players’ economic profits. Assuming that price and cost per unit of effort are constant, this is given by: 2.1 A Simple Non-cooperative Game

3. This can be represented by: The solution yields the players’ reaction functions: This is usually designated as an “inverse efficiency parameter”, as it increases with the cost per unit of effort and decreases with price and catchability coefficient. 2.1 A Simple Non-cooperative Game

4. Let us now look at the solution of the game. The Nash equilibrium is adopted – this is the most important solution concept for non-cooperative resource games. A Nash equilibrium occurs when each player chooses a strategy that maximizes his payoff given the other players’ strategies. Thus in the Nash equilibrium each player’s strategy is an optimal response to the other players’ strategies. In the Nash equilibrium no player has incentive to deviate unilaterally. 2.1 A Simple Non-cooperative Game

5. The Nash-equilibrium fishing effort strategies and the corresponding payoffs are given, respectively, by: The steady-state equilibrium stock level is the following: Note: it depends positively on the carrying capacity of the ecosystem and negatively on players’ efficiency. 2.1 A Simple Non-cooperative Game

6. Let us illustrate with a numerical example, using the following parameters: The players’ reaction functions are given by: The Nash equilibrium occurs at the interception of the two reaction functions: 2.2 The “Prisoner’s Dilemma” in Fisheries (cont.)

8. Most studies are centred on finding the equilibrium number and size of coalitions and share a common two-stage game framework. 3. Partition Function Games In the first stage players form coalitions, and in the second-stage coalitions engage in noncooperative behaviour. The coalition payoffs in the second stage are defined as a partition function. This function assigns a value to each coalition, which depends on the entire coalition structure.

9. Let us introduce a partition function game on a straddling stock fishery based on: Pintassilgo, P. and M. Lindroos (2007). Coalition Formation in Straddling Stock Fisheries: A Partition Function Approach. International Game Theory Review. 3.1 Coalition Formation in Straddling Stock Fisheries: A Partition Function Approach The paper models straddling stock fisheries through a game in partition function form using the classical Gordon-Schaefer bioeconomic model.

10. Partition Function The Per-member Partition Function was computed for all coalition structures: Full Cooperation The grand coalition solves the following problem: What is the solution of this problem ?

11. Partition Function Usually referred as “inverse efficiency parameter”. The stock level is: Each member of the grand coalition receives the following payoff:

12. Partition Function Non - Cooperation Consider a generic coalition structure with two or more coalitions:

13. Partition Function  Compute the reaction function of each coalition.  What is the fishing effort of each coalition at the Nash equilibrium ?  What is the equilibrium stock level ?  What are the coalitions’ payoffs ?

14. Stability and Equilibrium Coalition Structures A coalition structure is stand-alone stable if and only if no player finds it profitable to leave his coalition to form a singleton coalition, holding the rest of the coalition structure constant. According to Yi (1997), in the context of positive externalities the concept of stand-alone stability (or internal stability) is particularly useful, namely in characterizing equilibrium coalition structures. Result 1 In this fishery game, the grand coalition is stand-alone stable if and only if the number of players is two. Definition

15. Simulation Results Potential Internal Stability Likelihood

16. Partition Function Game