Non-cooperative game theory: Three fisheries games

1. Non-cooperative game theory: Three fisheries games Marko Lindroos JSS

2. This lecture is about • Non-cooperative games • classification • Nash equilibrium • Applications in fisheries economics • basic game (Mesterton-Gibbons NRM 1993) • stage games (Ruseski JEEM 1998) • repeated games (Hannesson JEEM 1997)

3. Non-cooperative games • Individual strategies for the players • Reaction functions, best reply • Nash equilibrium definition • Stages games at different levels • Repeated games, folk theorems, sustaining cooperative behaviour as equilibria • Dynamic games

4. Why non-cooperative • Classification: strategic (static), extensive (dynamic), coalition • Important in fisheries non-cooperation (competition) vs cooperation • Division not clear, almost all games have both non-cooperative and cooperative elements • Typically in economics non-cooperative game theory dominates

5. What are non-cooperative games about • How fisher’s decisions interact with other fishers’ decisions • What is the best strategy for the fishers • What is exected to happen is the fishery? Depends on rules of the game, number of players, biological factors • Why fishers behave as they do? • Assume rational choice

6. International fisheries negotiations Nature of negotiations • Countries attempt to sign and ratify agreements to maximise their own economic benefits • Negotiations typically time-consuming • Agreements not binding  self-enforcing or voluntary agreements

7. Explaining the tragedy of the commons • Can we explain the seemingly irrational behaviour in the world’s fisheries, overexploitation, overcapitalisation, bycatch… • Non-cooperative game theory explains this behaviour • Non-cooperative games vs open access (freedom of the seas)

8. Nash equilibrium • Each player chooses the best available decision • It is not optimal for any single player to unilaterally change his strategy • There can be a unique equilibrium, multiple equilibria or no equilibria

9. Fisher’s dilemma • Modified prisoner’s dilemma • Non-cooperation vs cooperation • Example 1: Two countries exploiting a common fish stock

10. Fisher’s dilemma explanation • Deplete: Corresponds to non-cooperation. The country is only interested in short-run maximisation of economic benefits. No regulation. • Conserve: Optimal management of the fishery. Cooperative case. • The cooperative solution (Conserve, Conserve) maximises the joint payoffs to the countries, equal to 50. However, neither of the countries is satisfied with the cooperative strategy. Both would gain by changing their strategy to Deplete (free-riding). This is the game-theoretic interpretation of tragedy of the commons. • In the Nash equilibrium (Deplete, Deplete) unilateral deviation is not optimal for the countries.

11. Reaction (best response) functions • Gives the best decisions a player can make as a function of other players’ decisions • If a decision is not a best response it can not be a Nash equilibrium • Typically best response functions are derived from a set of optimisation problems for the players. In an n player game there are n best response functions. • Nash equilibrium is found at the intersection of the best response functions (solution to the system of equations) • Strategy is best response if it is not strictly dominated

12. Repeated games • deterring short-term advantages by a threat or punishment  in the fisher’s dilemma escaping the non-cooperative Nash equilibrium • folk theorems (understood not published) • credibility of threats

13. Numerical repeated game • Assume that the game in example 1 is repeated infinite number of times. If one player deviates from the cooperative strategy Conserve to the non-cooperative strategy Deplete, it will also trigger the other player to choose Deplete forever after the deviation. This means that both countries punish severely deviations from the common agreement. • Cooperation can be sustainable if the present value of choosing Conserve is higher than deviating once from cooperation. • Present value of cooperation to player 1 when discount rate is 5%:

14. Cooperation vs. deviation • This infinite sum of the geometric progression and can be solved as follows: • = 600 • Next we calculate the present value of deviation. Country 1 first receives 40 and thereafter only 3 since country 2 uses its trigger strategy, according to which it never again signs an agreement. • Hence, the present value of deviation is: = 37 +3/(1-0.95) = 97

15. Tragedy of the commons solved • We see that the present value of deviating is clearly smaller and thus, cooperation (Conserve, Conserve) is now the equilibrium of the repeated game. • Note that the discount rate is critical in repeated games. As discount rate approaches infinity the present value of cooperation approaches 30 and the present value of deviation approaches 40. The critical discount rate, over which deviation is profitable, is therefore finite.

16. The first non-cooperative fisheries game • Assume there are n players (fishers, fishing firms, countries, groups of countries) harvesting a common fish resource x • Each player maximises her own economic gains from the resource by choosing a fishing effort Ei • This means that each player chooses her optimal e.g. number of fishing vessels taking into account how many the other players choose • As a result this game will end up in a Nash equilibrium where all individual fishing efforts are optimal

17. Building objective functions of the players • Assume a steady state: • By assuming logistic growth the steady state stock is then hi=qEix Stock biomass depends on all fishing efforts

18. Objective function • Players maximise their net revenues (revenues – costs) from the fishery • max phi –ciEi • Here p is the price per kg, hi is harvest of player i, ci is unit cost of effort of player i

19. Deriving reaction curves of the players • The first order condition for player i is • The reaction curve of player i is then bi=ci/pqK

20. Equilibrium fishing efforts • Derive by using the n reaction curves • The equilibrium fishing efforts depend on the efficiency of all players and the number of players

21. Illustration • Nash-Cournot equilibrium • Symmetric case • Schäfer-Gordon model

22. Exercises • Compute the symmetric 2-player and n player equilibrium. First solve 2-player game, then extend to n players.

23. A two-stage game (Ruseski JEEM 1998) • Assume two countries with a fishing fleet of size n1 and n2 • In the first stage countries choose their optimal fleet licensing policy, i.e., the number of fishing vessels. • In the second stage the fishermen compete, knowing how many fishermen to compete against • The model is solved backwards, first solving the second stage equilibrium fishing efforts • Second, the equilibrium fleet licensing policies are solved

24. Objective function of the fishermen • The previous steady state stock is then • The individual domestic fishing firm v maximises

25. Reaction functions • In this model the domestic fishermen compete against domestic vessels and foreign vessels • The reaction between the two fleets is derived from the first-order condition by applying symmetry of the vessels

26. Equilibrium fishing efforts • Analogously in the other country • By solving the system of two equations yields the equilibrium

27. Equilibrium stock • Insert equilibrium efforts into steady state stock expression • The stock now depends explicitly on the number of the total fishing fleet

28. Equilibrium rent • Insert equilibrium efforts and stock into objective function to yield

29. First stage • The countries maximise their welfare, that is, fishing fleet rents less management costs • The optimal fleet size can be calculated from the FOC (implicit reaction function)

30. Results • Aplying symmetry and changing variable m = 1+2n • With F=0  open access

31. Discussion • Subsidies • Quinn & Ruseski: asymmetric fishermen • entry deterring strategies: Choose large enough fleet so that the rival fleet is not able make profits from the fishery • Kronbak and Lindroos ERE 2006 4 stage coalition game

32. Repeated games – a step towards cooperation • When cooperation is sustained as an equilibrium in the game • The game is repeated many times (infinitely) • The players use trigger strategies as punishment if one of the players defects from the cooperative strategy • Trigger here means that defection triggers non-cooperative behaviour for the rest of the game • Cooperation means higher fish stock than non-cooperation, in the defection period the stock is between cooperative and non-cooperative levels

33. Cooperative strategies • Cooperative effort from SG-model • Cooperative fish stock • Cooperative benefits

34. Optimal defection effort • Best response when all others choose the cooperative strategy • Optimal defection effort

35. Non-cooperative strategies • Effort • Stock

36. Cooperation vs. cheating • Benefits from cheating • Condition for cooperative equilibrium

37. Discussion • Hannesson (JEEM 1997) similar results • Higher costs and lower discount rate enable a higher number of countries in the cooperative equilibrium • Self-enforcing agreements

38. On Species Preservation and Non-Cooperative Exploiters

39. Outline • Motivation • Model • Results • Conclusion • Discussion

40. Motivation • Combining two-species models with the game theory • What are the driving force for species extinction in a two-species model with biological dependency? • Does ‘Comedy of the Commons’ occur in two-species fisheries? • What are the ecosystem consequences of economic competition?

41. Modelling approach • Two-species • n symmetric competitive exploiters with non-selective harvesting technology • Fish stocks may be biologically independent or dependent • What is the critical number of exploiters?

42. Analytical independent species model • S-G model • Derive first E* as the optimal effort, it depends on the relevant economic and biological parameters • An n-player equilibrium is then derived as a function of E*and n. • Relate then the equilibrium to the weakest stock’s size to compute critical n*, over which ecosystem is not sustained.

43. Dependent vs independent species • Driving force of extinction: • Independent species • Biotechnical productivity • Economic parameters • Dependent species • Biological parameters must be considered • Gives rise to a complex set of conditions • For example: • Natural equilibrium does not exist • ‘The Comedy of the Commons’

44. Numerical dependent species model • Cases illustrated: Biological competition, symbiosis and predator-prey Case 1: Both stocks having low intrinsic growth rate Case 2: Both stocks having a high intrinsic growth rate Case 3: Low valued stock has a low intrinsic growth rate, high value stock has a high intrinsic growth rate. Case 4: Low valued stock has a high intrinsic growth rate, high value stock has a low intrinsic growth rate. Parameter values applied for simulation

45. Case 1: low intrinsic growth rate

46. Case 2: High growth

47. Case 4: Low valued stock has a high intrinsic growth rate, high value stock has a low intrinsic growth rate. Opposite case 3

48. Conclusion • ‘Tragedy of the Commons’ does not always apply • A small change in the interdependency can lead to big changes in the critical number of non-cooperative players • With competition among species a higher intrinsic growth rate tend to extend the range of parameters for which restricted open access is sustained

49. Discussion • From single-species models to ecosystem models • Ecosystem approach vs. socio-economic approach • Agreements and multi-species