Climate Economics: International Cooperation Fall 2024 Lecture Slides PDF

Climate Economics: International Cooperation Ralph Winkler Department of Economics and Oeschger Centre for Climate Change Research Universität Bern Fall Term 2024 R. Winkler (VWI & OCCR)...

Climate Economics: International Cooperation Ralph Winkler Department of Economics and Oeschger Centre for Climate Change Research Universität Bern Fall Term 2024 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –1– Organization Term: Fall Term 2024 Course: Climate Economics: International Cooperation Time and Place: Tue 12:15–16:00, 106 H4 (Lecture) Wed 8:15–10:00, 106 H4 (Tutorial bi-weekly) 25 Sep / 9 Oct / 23 Oct / 6 Nov / 20 Nov / 4 Dec Lectures Ralph Winkler & Tutorials: A 217 UniS, [email protected] Office Hours: by appointment Course Type: MA Economics, MA Climate Science ECTS: 4.5 Examination: Take-Home Exam (Deadline: 31 Jan 2025) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –2– Preliminaries Profound knowledge in microeconomics and non-cooperative game theory Microeconomics II Einführung in die Spieltheorie Basic knowledge in environmental economics recommended Environmental Economics: Introduction R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –3– Relations to other Courses Climate Economics Space Time global public good long time horizon no supranational scientific uncertainty authority Climate Economics: Climate Economics: International Cooperation Scientific & Economic Foundations R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –4–  1 Introduction R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –5– 1.1 Anthropogenic Climate Change in a Nutshell R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –6– Anthropogenic Climate Change in a Nutshell Empirical Evidence: Greenhouse Gas Concentrations Atmospheric concentration of carbon dioxide, methane and nitrous oxide over the last 10,000 years (large panels) and since 1750 (inset panels) Source: IPCC (2007) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –7– Anthropogenic Climate Change in a Nutshell Empirical Evidence: Global Mean Temperature Global mean surface temperature from 1850–2005 as deviation from the mean surface temperature over the period from 1961–1990 Source: IPCC (2007) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –8– Anthropogenic Climate Change in a Nutshell Forecasts: Global Impacts of Climate Change (i) Global perspective on climate-related risks. Undetectable risk (white) indicates no associated impacts are detectable and attributable to climate change. Moderate risk (yellow) indicates that associated impacts are both detectable and attributable to climate change with at least medium confidence. High risk (red) indicates severe and widespread impacts, and purple shows that very high risk is indicated for key risks. Source: IPCC (2014a) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 –9– Anthropogenic Climate Change in a Nutshell Forecasts: Global Impacts of Climate Change (ii) Global mean temperature changes up to 2 °C (above 1986–2005 levels) would exacerbate current key impacts, such as loss of glaciers, and increases in the frequency and/or intensity of extreme events (high confidence), and trigger others, such as reduced food security in many low latitude nations (medium confidence). Global mean temperature changes of 2–4 °C would result in increasing number of key impacts at all scales (high confidence), such as widespread loss of biodiversity, decreasing global agricultural productivity and widespread deglaciation of Greenland (high confidence) and West Antarctic (medium confidence) ice sheets. Global mean temperature changes greater than 4 °C would lead to major increases in vulnerability (very high confidence), exceeding the adaptive capacity of many systems (very high confidence). R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 10 – Anthropogenic Climate Change in a Nutshell Forecasts: Climate Change in Switzerland in 2050 Until 2050 mean temperature in winter will increase by 1.8 °C and in summer by 2.7 °C Precipitation will increase in winter and decrease in summer More frequently rain instead of snow in winter, 75% of water stored in glaciers will be lost Increasing probability of extreme weather events such as heat waves, extreme rainfall and spring floodings Increase of annual mean temperature between 2–3 °C has positive effect on agriculture but higher variability due to weather extremes and reduced summer precipitation Skiing tourism will be negatively affected Less energy consumption for heating but increased energy consumption for cooling R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 11 – Anthropogenic Climate Change in a Nutshell Forecasts: Climate Change in Switzerland: Temperature & Precipitation Change in temperature (above) and in rainfall (below) in 2050 compared to 1990 (blue numbers: median; red numbers: 95% confi- dence interval). Source: OcCC (2007) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 12 – Anthropogenic Climate Change in a Nutshell Forecasts: Climate change in Switzerland – Temperature Distribution Summer temperatures (JJA) for Switzerland observed during 1864 to 2003 (top), simulated using a regional climate model for the period 1961 to 1990 (middle), and simulated for 2071 to 2100 under the 2 scenario (bottom). Source: IPCC (2007) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 13 –  1.2 International Climate Policy R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 14 – International Climate Policy The Science of Global Climate Change... Remaining carbon budget in GtCO2 from 2020 onwards (IPCC 2021): Probability to reach goal 17% 33% 50% 67% 83% 1.5° C 900 650 500 400 300 ∆T 1.7° C 1450 1050 850 700 550 2.0° C 2300 1700 1350 1150 900 Current yearly CO2 emissions: approx. 40 GtCO2 At current CO2 emission rates this implies: 9 years left for 1.5° C target at likelihood of 50% 17 years left for 1.7° C target at likelihood of 50% 30 years left for 2.0° C target at likelihood of 50% R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 15 – International Climate Policy... and the Politics of Global Climate Change 1 1992: UNFCCC – Rio de Janeiro, Brazil The UNFCCC takes measures to... stabilize greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system... 2 1997: Kyoto Protocol – Kyoto, Japan Commitment of developed world to reduce GHG emissions between 2008 and 2012 by roughly 5% against 1990 levels 3 2009: Copenhagen Accord – Copenhagen, Denmark The UNFCCC recognizes... the scientific view that the increase in global temperature should be below 2 °C... 4 2015: Paris Agreement – Paris, France The UNFCCC aims at... holding the increase in the global average temperature to well below 2 °C above pre-industrial levels and to pursue efforts to limit the temperature increase to 1.5 °C above pre-industrial levels... R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 16 – International Climate Policy The Climate Puzzle Fundamental Scientific and Political Consensus Drastically reduced greenhouse gas emissions compared to business-as-usual would substantially improve global welfare. Then, why do we not see drastic GHG emission reductions? R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 17 – International Climate Policy Obstacles for Effective Climate Policy 1 GHG emissions are global pollutant → Atmospheric GHG concentrations depend on global GHG emissions → Environmental damage depends on cumulative global GHG emissions 2 No supranational authority → Only voluntary international cooperation → International agreements must be self-enforcing 3 Heterogeneity of countries → Environmental damages / abatement costs → History and prospects of GHG emissions 4 Long time horizon and uncertainty → Raises concerns of intergenerational equity → How to deal with irresolvable uncertainty? R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 18 –  1.3 Two Introductional Experiments R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 19 – Introduction Experiment 1: The Ultimatum Game The Ultimatum Game is about the distribution of a certain asset Two players shall distribute 10 CHF among them Player 1 suggests a distribution (lets say in integer steps) Player 2 can accept (A) player 1’s distribution or decline (D) the suggestion If Player 2 accepts, the asset is distributed accordingly, if player 2 declines both players get nothing R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 20 – Introduction Experiment 2: Divide and Choose The game Divide and Choose is also about the distribution of a certain asset Two players shall distribute 10 CHF among them Player 1 divides the asset into two parts (lets again say in integer steps) Player 2 chooses one of the parts Player 2 gets the chosen part, player 1 gets the other part R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 21 – Table of Contents: Part I – Preliminaries 1 Introduction Anthropogenic Climate Change in a Nutshell International Climate Policy Two Introductional Experiments Literature 2 Non-cooperative Game Theory in a Nutshell Normal-Form Games and Nash Equilibrium Extensive-Form Games Subgame Perfect Equilibrium Literature R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 22 – Table of Contents: Part II – Current real world affairs 3 Global GHG Emissions as a Non-Cooperative Game A Simple GHG Model The Global Social Optimum The Global Emissions Game Comparison of Global Emissions Game and Global Social Optimum 4 International Cooperation as Non-Cooperative Game The Coalition Formation Game Modest International Cooperation Literature 5 International Permit Markets International Permit Markets as a Game Permit Market Equilibrium Permit Supply Initial Participation Numerical Illustration Literature R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 23 – Table of Contents: Part III – New Mechanisms 6 Designing International Environmental Agreements Desiderata for Optimal Treaty Design A Simple Deposit-Based Mechanism Numerical Illustration Literature 7 International Permit Markets and Refunding Incentives for Non-Compliance A Permit Market Refunding Scheme Complete Auctioning Partial Auctioning Membership Choice Literature 8 A Deposit-Refunding Mechanism A Generic Deposit-Refunding Mechanism An Abatement Contest Emission Levels Participation Minimum Deposit Literature R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 24 –  1.4 Literature R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 25 – Literature for Section 1 IPCC (2007): Climate Change 2007. The Physical Science Basis. http://www.ipcc.ch/report/ar4/wg1/ IPCC (2013): Climate Change 2013: The Physical Science Basis. http://www.ipcc.ch/report/ar5/wg1/ IPCC (2014a): Climate Change 2014: Impacts, Adaptation, and Vulnerability. http://www.ipcc.ch/report/ar5/wg2/ IPCC (2014b): Climate Change 2014: Mitigation of Climate Change. http://www.ipcc.ch/report/ar5/wg3/ IPCC (2021): Climate Change 2021: The Physical Science Basis. https://www.ipcc.ch/report/ar6/wg1/ OcCC (2007): Climate Change in Switzerland 2050. http://proclimweb.scnat.ch/products/CH2050/CH2050-report.html R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 26 – 2 Non-cooperative Game Theory in a Nutshell R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 27 – Non-cooperative Game Theory in a Nutshell What is Game Theory? Game Theory analyzes decision contexts in which two or more agents (players) interact strategically Strategic Interaction: Decision of one player has consequences not only for respective player but also for other players Term stems from analyzing popular parlor games like chess, checkers, nine men’s morris etc., which also constitute games in game theoretic sense: two or more players in interactive decision contexts Game Theory would be better termed “Interactive Decision Theory” R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 28 – 2.1 Normal-Form Games and Nash Equilibrium R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 29 – Normal-Form Games and Nash Equilibrium Characterization of Games Games in the game-theoretic sense are characterized by following properties: 1 Set of players 2 A characterization of all feasible strategies for all players 3 A characterization of players’ information during each stage of the game 4 A characterization of consequences chosen strategies imply for players 5 A characterization of players’ preferences over all possible outcomes of the game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 30 – Normal-Form Games and Nash Equilibrium Notation Notation to characterize games: Set of players: N = 1,... , n Set of feasible strategies of player i: Si Denote a feasible strategy of player i by si ∈ Si s−i = (s1 ,... , si−1 , si+1 ,... , sn ) is vector of strategies of all players except player i Vector s = (s1 ,... , sn ) = (si , s−i ) is a strategy profile Let S = S1 × · · · × Sn and S−i = S1 × · · · × Si−1 × Si+1 × · · · × Sn Pay-off function ui (s) characterizes consequences for player i depending on the strategy profile s R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 31 – Normal-Form Games and Nash Equilibrium Individual Rationality Assumption (Individual Rationality) All players act rational, i.e. they choose their strategies such as to maximize (expected) value of their own pay-off function. Rationality does not necessarily imply that players are selfish Pay-offs in Game Theory are not necessarily monetary payments In particular, pay-offs may depend on the well-being of other players or some fairness criteria R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 32 – Normal-Form Games and Nash Equilibrium Static Games of Complete Information Definition (Static Game) A Game is static if all players choose their strategy simultaneously, i.e. without knowing the strategy of all other players. Definition (Complete Information) Complete Information implies that all strategy sets Si and all pay-off functions ui (s) for i = 1,... , n are common knowledge. Definition (Common Knowledge) A property of a game is common knowledge if it is known to all players, and all players know that it is known to all players, and that all players know that all players know that it is known to all players, etc. R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 33 – Normal-Form Games and Nash Equilibrium Definition Normal-Form The Normal-Form presentation of games is particularly suited for the analysis of static games: Definition (Normal-Form Game) A Normal-Form Game is characterized by the set of players N , the sets of players’ strategies S1 ,... , Sn , and the players’ pay-off functions u1 (s),... , un (s). We denote such a game by Γ = {S1 ,... , Sn ; u1 (s),... , un (s)} We call Γ a Normal-Form Game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 34 – Normal-Form Games and Nash Equilibrium Matrix Representation for 2-Player Normal-From Games 2-Player Normal-From Games with finite strategy sets for both players can be represented by a matrix Let |S1 | = k, |S2 | = l and sij denotes the j-th element of strategy set Si of player i Player 2 s21 ··· s2l s11 (u1 (s11 , s21 ), u2 (s11 , s21 )) ··· (u1 (s11 , s2l ), u2 (s11 , s2l )) Player 1............ s1k (u1 (s1k , s21 ), u2 (s1k , s21 )) ··· (u1 (s1k , s2l ), u2 (s1k , s2l )) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 35 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma (i) Two identical countries can either abate GHG emissions (A) or continue to pollute (P) Abating GHG emissions induces costs of 5 billion CHF, polluting has no costs Benefits from emission abatement depend on how many countries abate: 0 CHF, if no country abates 4 billion CHF, if only one country abates 7 billion CHF, if both countries abate R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 36 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma (ii) The Prisoners’ Dilemma is a static game with complete information Prisoners’ Dilemma as a normal-form game: Country 2 A P A (2, 2) (−1, 4) Country 1 P (4, −1) (0, 0) Countries choose their feasible action (P) or (A) “simultaneously” , i.e. without knowing the other’s action Which actions will countries choose if they maximize their individual net benefits from GHG emission abatement? R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 37 – Normal-Form Games and Nash Equilibrium Nash Equilibrium To solve the Prisoners’ Dilemma we introduce a solution concept called Nash equilibrium (Nash 1950) The fundamental idea of the Nash equilibrium is that all players choose their best strategy given the strategies of all other players As a consequence, in a Nash equilibrium individual defection of any player is not profitable R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 38 – Normal-Form Games and Nash Equilibrium Definition: Best Response We seek each player’s best strategy, given the strategies of all other players This strategy is called the best response to the given strategy profile of all other players Definition (Best Response) A strategy si of player i is a best response to a given strategy profile s−i of all other players , if it maximizes player i’s pay-off: ui (si , s−i ) ≥ ui (s′i , s−i ) , ∀ s′i ∈ Si (BR) The set of all best responses of player i to a given strategy profile s−i of all other players is denoted by Ri (s−i ). R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 39 – Normal-Form Games and Nash Equilibrium Definition: Nash Equilibrium A strategy profile is a Nash equilibrium, if it is the mutual best response of all players Definition (Nash Equilibrium) A strategy profile s⋆ = (s⋆1 ,... , s⋆n ) is a Nash equilibrium (NE), if for all players i = 1,... , n the strategy s⋆i of player i is the best response to the strategy profile s⋆−i = (s⋆1 ,... , s⋆i−1 , s⋆i+1 ,... , s⋆n ) of all other players, i.e. ui (s⋆i , s⋆−i ) ≥ ui (s′i , s⋆−i ) , ∀ s′i ∈ Si ∀ i = 1,... , n As a consequence: s⋆i = arg max ui (si , s⋆−i ) , ∀ i = 1,... , n si ∈Si R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 40 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma Back to the Prisoners’ Dilemma: Country 2 A P A (2, 2) (−1, 4) Country 1 P (4, −1) (0, 0) We find the NE In fact, polluting (P) is dominant strategy for both countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 41 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma Back to the Prisoners’ Dilemma: Country 2 A P A (2, 2) (−1, 4) Country 1 P (4, −1) (0, 0) We find the NE In fact, polluting (P) is dominant strategy for both countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 41 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma Back to the Prisoners’ Dilemma: Country 2 A P A (2, 2) (−1, 4) Country 1 P (4, −1) (0, 0) We find the NE In fact, polluting (P) is dominant strategy for both countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 41 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma Back to the Prisoners’ Dilemma: Country 2 A P A (2, 2) (−1, 4) Country 1 P (4, −1) (0, 0) We find the NE In fact, polluting (P) is dominant strategy for both countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 41 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma Back to the Prisoners’ Dilemma: Country 2 A P A (2, 2) (−1, 4) Country 1 P (4, −1) (0, 0) We find the NE → (P,P) In fact, polluting (P) is dominant strategy for both countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 41 – Normal-Form Games and Nash Equilibrium Example: The Prisoners’ Dilemma Back to the Prisoners’ Dilemma: Country 2 A P A (2, 2) (−1, 4) Country 1 P (4, −1) (0, 0) We find the NE → (P,P) In fact, polluting (P) is dominant strategy for both countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 41 –  2.2 Extensive-Form Games R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 42 – Extensive-Form Games Motivation Normal form does not account for dynamic development of games Obviously, this is inconsequential for static games But many games exhibit dynamic and sequential structure: players move sequentially and can at least partly observe what other players have done so far We know investigate dynamic games with complete information R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 43 – Extensive-Form Games Definition: Extensive-Form Game Dynamic games are best analyzed in extensive form Definition (Extensive-Form Game) An extensive-form game G is characterized by: 1 Set of players 2 Timing of events: which player moves when during the game 3 All feasible actions of all players for all their moves 4 All players’ information about the past history of the game when they move 5 Pay-off functions of all players for all feasible outcomes of the game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 44 – Extensive-Form Games Game Tree Extensive-form games are best represented by a game tree This is a connected and loopless graph Players’ moves are represented by decision nodes Feasible actions at each decision node are represented by edges The game tree starts with the decision node of the player who moves first and...... ends at the terminal nodes, representing all possible outcomes of the game Thus, every terminal node corresponds to a pay-off profile for all players R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 45 – Extensive-Form Games Example: The Sequential Prisoners’ Dilemma (i) Consider the Sequential Prisoners’ Dilemma Actions and pay-offs are identical to the Prisoner’s Dilemma, except that Country 2 observes Country 1’s action before it acts itself Game tree: Country 1 b A P b b Country 2 A P A P (2, 2) (−1, 4) (4, −1) (0, 0) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 46 – Extensive-Form Games Definition: Strategy In static games of complete information actions and strategies of players always identical Yet, a strategy in the game theoretic sense is complete description of all the actions a player chooses during a game Definition (Strategy) A players’ strategy specifies the respective player’s actions in all his decision nodes of a game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 47 – Extensive-Form Games Example: The Sequential Prisoners’ Dilemma (ii) In the Sequential Prisoners’ Dilemma Country 1 has 2 feasible strategies, which are identical to its feasible actions when it has the move: 1 Abate: A 2 Pollute: P Country 2 has also 2 feasible actions when it has the move, but 4 feasible strategies: 1 Abate, if country 1 abates, and abate, if country 1 pollutes: AA 2 Pollute, if country 1 abates, and abate, if country 1 pollutes: PA 3 Abate, if country 1 abates, and pollute, if country 1 pollutes: AP 4 Pollute, if country 1 abates, and pollute, if country 1 pollutes: PP R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 48 – Extensive-Form Games Dynamic Games in Normal Form Once, all players’ strategy sets of a dynamic game are characterized, we can represent it in normal form: The Sequential Prisonners’ Dilemma in normal form: Country 2 AA PA AP PP A (2, 2) (−1, 4) (2, 2) (−1, 4) Country 1 P (4, −1) (4, −1) (0, 0) (0, 0) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 49 – Extensive-Form Games Nash Equilibria of Dynamic Games Now we can also determine the NEs of the Sequential Prisoners’ Dilemma: Country 2 AA PA AP PP A (2, 2) (−1, 4) (2, 2) (−1, 4) Country 1 P (4, −1) (4, −1) (0, 0) (0, 0) The Sequential Prisoner’s Dilemma exhibits a unique NE (P,PP) in which both countries always pollute R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 50 –  2.3 Subgame Perfect Equilibrium R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 51 – Subgame Perfect Equilibrium Introductory Example: The Ultimatum Game (i) Recall the Ultimatum Game from the introductory experiment We can interpret the Ultimatum Game as a dynamic game: 1 Player 1 proposes a division of the asset 2 Player 2 accepts (A) or declines (D) player 1’s proposal Pay-offs: If player 2 accepts, pay-offs are as proposed by player 1 If player 2 declines, both players receive a pay-off of 0 Tie-breaking rule: If indifferent, player 2 accepts proposal R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 52 – Subgame Perfect Equilibrium Introductory Example: The Ultimatum Game (ii) Game tree of the Simplified Ultimatum Game: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 53 – Subgame Perfect Equilibrium Introductory Example: The Ultimatum Game (iii) Strategies: 1 Player 1: Offer (0), (1), (3) 2 Player 2: 3-tuple XXX (action for (0), action for (1) and action for (3)) AAA, AAD, ADA, DAA, ADD, DAD, DDA, DDD Simplified Ultimatum game in normal form : Player 2 AAA AAD ADA DAA ADD DAD DDA DDD 0 (10, 0) (10, 0) (10, 0) (0, 0) (10, 0) (0, 0) (0, 0) (0, 0) Player 1 1 (9, 1) (9, 1) (0, 0) (9, 1) (0, 0) (9, 1) (0, 0) (0, 0) 3 (7, 3) (0, 0) (7, 3) (7, 3) (0, 0) (0, 0) (7, 3) (0, 0) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 54 – Subgame Perfect Equilibrium Introductory Example: The Ultimatum Game (iv) Simplified Ultimatum Game has eight NEs: (0,AAA), (0, AAD), (0, ADA), (0,ADD), (0,DDD), (1,DAA), (1,DAD), (3,DDA) Our tie-breaking rule eliminates one NE: (0,DDD) Out of the remaining 7 NEs some seem implausible: Consider, e.g., (3,DDA): NE because player 2 “threatens” to decline offers of 0 and 1 But is this “threat” credible? Suppose player 1 had offered 1 Player 2 gets a pay-off of 1 when playing (A) and 0 when playing (D) Thus, playing (D) for an offer of 1 is not rational Anticipating rational behavior, player 1 should not give into implausible threats R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 55 – Subgame Perfect Equilibrium Motivation We have seen that dynamic games may exhibit NE that are irrational: 1 NE is supported by threat to play a certain action ai in response to actions a−i of all other players 2 But: given a−i has been played, action ai does not maximize player i’s pay-off Threat to play ai in response to actions a−i is, thus, implausible (as long as player i is rational) We introduce a refinement of the NE, the so called subgame perfect equilibrium (SPE), which eliminates NEs based on implausible threats R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 56 – Subgame Perfect Equilibrium Subgames and Subgame Perfect Equilibrium First, we define a subgame Definition (Subgame) A subgame of an extensive-form game G is a subset of the game with the following properties: 1 It starts with a decision node 2 It contains all subsequent decision and terminal nodes (and only these) Now, we can introduce the subgame perfect equilibrium: Definition (Subgame Perfect Equilibrium (SPE)) A strategy profile s = (s1 ,... , sn ) of an extensive-form game G is a subgame perfect equilibrium (SPE) if it induces a Nash equilibrium in every subgame. Set of SPEs of an extensive-form game is subset of its set of NEs R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 57 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (i) Recall the Simplified Ultimatum Game: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) The game has 4 subgames R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 58 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (i) Recall the Simplified Ultimatum Game: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) The game has 4 subgames R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 58 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (i) Recall the Simplified Ultimatum Game: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) The game has 4 subgames R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 58 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (i) Recall the Simplified Ultimatum Game: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) The game has 4 subgames R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 58 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (i) Recall the Simplified Ultimatum Game: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) The game has 4 subgames R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 58 – Subgame Perfect Equilibrium Backward Induction We find SPEs of an extensive-from game by backward induction: We start at the last decision nodes of the game and determine the best action of the players which have the move We move to the but last decision nodes of the game and determine the best actions of the players having the move, given the just determined actions on the last decision node We follow this procedure to the first decision node The set of determined best actions are SPEs of the game Backward induction solves a game literally from the end to the beginning R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 59 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (ii) We solve the Simplified Ultimatum Game by backward induction: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) Thus, the unique SPEClimate R. Winkler (VWI & OCCR) is (0,AAA) Economics: Intl. Cooperation Fall Term 2024 – 60 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (ii) We solve the Simplified Ultimatum Game by backward induction: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) Thus, the unique SPEClimate R. Winkler (VWI & OCCR) is (0,AAA) Economics: Intl. Cooperation Fall Term 2024 – 60 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (ii) We solve the Simplified Ultimatum Game by backward induction: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) Thus, the unique SPEClimate R. Winkler (VWI & OCCR) is (0,AAA) Economics: Intl. Cooperation Fall Term 2024 – 60 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (ii) We solve the Simplified Ultimatum Game by backward induction: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) Thus, the unique SPE is (0,AAA) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 60 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (ii) We solve the Simplified Ultimatum Game by backward induction: Player 1 b 0 1 3 Player 2 b b b A D A D A D (10, 0) (0, 0) (9, 1) (0, 0) (7, 3) (0, 0) Thus, the unique SPE is (0,AAA) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 60 – Subgame Perfect Equilibrium Example (cont.): The Ultimatum Game (ii) We solve the Simplified Ultimatum Game by backward induction: Thus, the unique SPE is (0,AAA) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 60 – Remark Experimental Evidence on the Ultimatum Game We just learned that (depending on the tie-breaking rule) the only SPE of the Ultimatum Game is (0,A... A) or (1,DA... A) Why do we observe very different results in experimental studies? Do people consistently behave irrational? No! Our analysis assumed that players’ pay-off functions are given by the monetary gains of the game But players’ pay-offs may also depend on other concerns: envy, greed, fairness, etc. R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 61 – Remark Experimental Evidence on the Ultimatum Game We just learned that (depending on the tie-breaking rule) the only SPE of the Ultimatum Game is (0,A... A) or (1,DA... A) Why do we observe very different results in experimental studies? Do people consistently behave irrational? No! Our analysis assumed that players’ pay-off functions are given by the monetary gains of the game But players’ pay-offs may also depend on other concerns: envy, greed, fairness, etc. R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 61 –  2.4 Literature R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 62 – Literature for Section 2 R. Gibbons (1992): Game Theory for Applied Economists. Princeton University Press. F. Munoz-Garcia & D. Toro-Gonzalez (2016): Strategy and Game Theory. Springer M. Osborne (2003): An Introduction to Game Theory. Oxford University Press M. Osborne & A. Rubinstein (1994): A Course in Game Theory. MIT Press. D. Fudenberg & J. Tirole (1991): Game Theory. MIT Press. R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 63 – 3 Global GHG Emissions as a Non-Cooperative Game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 64 –  3.1 A Simple GHG Model R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 65 – A Simple GHG Model Model Outline (i) World consists of n countries i = 1,... , n Country i’s receives country specific benefits Bi (ei ) from its GHG emissions ei : Bi (0) = 0 , Bi′ > 0 , Bi′′ < 0 , ∀ ei ∈ [0, ϵi ] ϵi denotes business-as-usual (BAU) emissions of country i R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 66 – A Simple GHG Model Model Outline (ii) GHG emissions of all countries accumulate to global emissions E: n X E= ei i=1 Country i suffers from country specific damages D(E) depending on global emissions E: Di (0) = 0 , Di′ > 0 , Di′′ ≥ 0 , ∀E>0 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 67 – A Simple GHG Model Model Outline (iii) Social welfare of country i is given by: Wi (ei , E) = Bi (ei ) − Di (E) We assume a benevolent government in all countries behaving such as to maximize its own country’s welfare R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 68 – A Simple GHG Model Functional Forms (i) We use the following functional forms: 1   Bi (ei ) = αi ei ϵi − ei 2 Di (E) = βi E Thus, we obtain for the first- and second-order derivatives: Bi′ (ei ) = αi (ϵi − ei ) , Di′ = βi Bi′′ = −αi , Di′′ = 0 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 69 – A Simple GHG Model Functional Forms (ii) We introduce the following abbreviations for later reference: n X n X B= βi , E= ϵi i=1 i=1 In addition, we impose the following assumption: αi ϵi > B , ∀ i = 1,... , n marginal benefits for ei = 0 higher than sum of marginal damages R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 70 – A Simple GHG Model Graphical Illustration Bi (ei ) Di (E) B1 (e1 ) B2 (e2 ) D1 (E) D2 (E) ǫ1 ǫ2 ei E R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 71 – A Simple GHG Model Economic Interpretation: Benefit Function The benefit function Bi (ei ) of country i characterizes productive capacity can be interpreted as GDP as a function of GHG emissions We define carbon efficiency: amount of GDP produced per unit of emissions In general, highly developed countries exhibit high carbon efficiency A high carbon efficiency also implies high emission abatement costs R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 72 – A Simple GHG Model Economic Interpretation: Damage Function The environmental damage function Di (E) of country i characterizes monetarized damages from anthropogenic climate change can be interpreted as willingness-to-pay as a function of GHG emissions Willingness-to-pay (WTP): Amount of money one is happy to sacrifice in order to avoid the damage In general, more developed countries exhibit higher WTPs for GHG emission abatement than less developed countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 73 – A Simple GHG Model Reality Check Does our simple GHG model capture essential features of global climate change? 1 GHG emissions are global pollutant ✓ 2 No supranational authority ✓ 3 Heterogeneity of countries ✓ 4 Long time horizon and uncertainty Highly stylized model omitting lots of important characteristics: (economic) cross-country relationships (e.g. trade, international relations, etc.) internal structure of countries (e.g. inequality, political system, etc.)... R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 74 –  3.2 The Global Social Optimum R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 75 – The Global Social Optimum Global Social Welfare Suppose a global social planner trying to determine optimal GHG emissions by maximizing global social welfare We define global social welfare as the sum of all countries’ social welfare: n n W GSO = X X Wi (ei , E) = [Bi (ei ) − Di (E)] i=1 i=1 Note that our global social welfare is strongly related to utilitarian concepts of welfare and the Kaldor-Hicks criterion in social choice theory R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 76 – The Global Social Optimum The Maximization Problem Formally, in Global Social Optimum (GSO) the social planner solves: n X max n [Bi (ei ) − Di (E)] {ei }i=1 i=1 subject to n X E= ei , ei ∈ [0, ϵi ] , ∀ i = 1,... , n i=1 Objective function strictly concave → local extremum (if it exists) is global welfare maximum R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 77 – The Global Social Optimum First-Order Conditions Necessary (and sufficient) conditions for interior global maximum: n Bi′ (ei ) Di′ (E) , X = ∀ i = 1,... , n i=1 Identical right-hand side for all countries i implies: Bi′ (ei ) = Bj′ (ej ) , ∀ i, j = 1,... , n Equi-marginal principle: In GSO marginal benefits equal across countries R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 78 – The Global Social Optimum Global Social Optimum Simultaneous solution of necessary conditions implies global social n optimum e⋆i i=1  For the assumed functional forms, the first-order conditions yield: αi (ϵi − ei ) = B , ∀ i = 1,... , n This yields for emissions in GSO: B e⋆i = ϵi − , ∀ i = 1,... , n αi n n ⋆ B e⋆i X X E = =E− i=1 i=1 αi R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 79 –  3.3 The Global Emissions Game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 80 – The Global Emissions Game Translation of Simple GHG Model into a Static Game We now translate simple GHG model into static non-cooperative game Global Emissions Game (GEG) is normal-form game characterized by: 1 Players: Countries i = 1,... , n 2 Strategy Sets: Emission levels ei ∈ [0, ϵi ] 3 Pay-off functions: We assume that countries only consider own welfare Wi (ei , E) = Bi (ei ) − Di (E) Additional assumption: all players choose emission levels simultaneously R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 81 – The Global Emissions Game Additional Definitions e = {e1 ,... , en } denotes a strategy profile The vector of strategies of all players j but player i is given by: e−i = {e1 ,... , ei−1 , ei+1 ,... , en } The sum of all but player i’s emissions is given by: X E−i = ej j̸=i R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 82 – The Global Emissions Game Best Response (i) To determine the NE of the GEG we first derive country i’s best response to given strategies e−i of all other players: Best response of country i is solution to: max Wi (ei , e−i ) = max Bi (ei ) − Di (ei + E−i ) ei ei Note that best response only depends on E−i and not e−i ! Objective function strictly concave → unique best response for given e−i , respectively E−i → Best response/Reaction function Ri (E−i ) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 83 – The Global Emissions Game Best Response (ii) Reaction functions Ri (E−i ) implicitly given by first-order conditions: F OC = Bi′ (ei ) − Di′ (ei + E−i ) = 0 , i = 1,... , n How does best response depend on emission choices of all other countries? From implicit function theorem: dRi (E−i ) Di′′ (ei + E−i ) = ′′ ≤0 dE−i Bi (ei ) − Di′′ (ei + E−i ) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 84 – The Global Emissions Game Best Response (iii) Thus, emission choices of countries are either strategic substitutes if dRi (E−i )/DE−i < 0, or dominant strategies if dRi (E−i )/DE−i = 0 Strategic substitutes: best for country i to increase own emissions if all other countries reduce their emissions Dominant strategies: best response independent of other countries’ emission choices For our functional forms best responses are dominant strategies: βi Ri (E−i ) = ϵi − αi R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 85 – The Global Emissions Game Nash Equilibrium (i) n Mutual best responses characterize Nash equilibrium (NE) êi  i=1 In case of dominant strategies, NE identical to best responses: βi êi = Ri (E−i ) = ϵi − , ∀ i = 1,... , n αi For general case, rewrite first-order conditions: ei = Bi′−1 Di′ (E) ,
i = 1,... , n Note: inverse function Bi′−1 exists, as Bi′ is strictly monotonic, as Bi′′ < 0 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 86 – The Global Emissions Game Nash Equilibrium (ii) Summing-up over all countries i = 1,... , n yields: n Bi′−1 Di′ (E) X
E= i=1 This equation determines global emission Ê in NE It yields unique solution for Ê, as LHS monotonically increasing and RHS monotonically decreasing in E n Inserting Ê back into first-order conditions yields unique NE êi  i=1 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 87 – 3.4 Comparison of Global Emissions Game and Global Social Optimum R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 88 – Comparison of GEG and GSO Emissions (i) Emission levels in Global social optimum (GSO): n B X B e⋆i = ϵi − , ∀ i = 1,... , n , E⋆ = E − αi α i=1 i Global Emissions Game (GEG): n βi X βi êi = ϵi − , ∀ i = 1,... , n , Ê = E − αi α i=1 i Individual and global emission levels lower in GSO than in GEG R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 89 – Comparison of NE and Global Social Optimum Emissions (ii) To understand underlying causes recall first-order conditions: n Bi′ (ei ) Dj′ (E) , X GSO: = ∀ i = 1,... , n j=1 GEG: Bi′ (ei ) = Di′ (E) , ∀ i = 1,... , n Emissions of country i not only affect country i but all countries → externality Externality each country imposes on all other countries accounted for in GSO but not in GEG R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 90 – Comparison of GEG and GSO Pareto Efficiency (i) At least for sufficiently symmetric countries all countries better off in GSO → GEG not Pareto efficient (Prisoner’s Dilemma) For heterogeneous countries GSO not necessarily Pareto improvement over GEG: Suppose country i exhibits βi = 0 In GEG country i plays êi = ϵi and receives welfare Ŵi = Bi (ϵi ) Country i’s emission level in GSO is e⋆i < ϵi implying Wi⋆ = Bi (e⋆i ) < Ŵi R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 91 – Comparison of GEG and GSO Pareto Efficiency (ii) If welfare transferable between countries → transferable utility (TU) game...... there exists transfer scheme: n X ti = 0 i=1 such that GSO plus transfer scheme is Pareto improvement Agreement of transfer scheme may be difficult in practice, as some countries receive transfers (ti > 0) while other countries pay transfers (ti < 0) → fairness considerations R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 92 – Comparison of GEG and GSO Global Social Welfare No matter whether GSO is Pareto improvement over GEG → global social welfare lower in GEG compared to GSO Why does GEG fall short of GSO? 1 Maximizing domestic welfare countries do not take into account negative externalities emissions impose on other countries → global emissions too high 2 Maximizing domestic welfare does, in general, not lead to equal marginal benefits across countries → wrong distribution of global emissions R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 93 – Comparison of GEG and GSO Plausibility of Global Emissions Game Is outcome of Global Emissions Game plausible? Result hinges on selfish pay-off functions: Do countries really only consider their own welfare/well-being? At least in good approximation: Yes Ample empirical evidence: e.g. foreign aid spendings But: if countries understand that GEG is prisoner’s dilemma is there no way to overcome problem? Yes, there is: International Agreements R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 94 – 4 International Cooperation as Non-Cooperative Game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 95 – International Cooperation as Non-Cooperative Game Motivation We have seen: Global Emissions Game is prisoners’s dilemma Countries have self-interest to improve on NE of Global Emissions Game Countries could try to cooperate in emissions reductions One possibility is Coalition Formation Game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 96 –  4.1 The Coalition Formation Game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 97 – The Coalition Formation Game Structure of the Game Let C be the set of countries in the coalition and k = |C| the number of coalition members Coalition Formation Game (CFG) is dynamic (2-stage) game: 1 Countries simultaneously decide whether to join coalition 2 Countries simultaneously decide about emissions: Coalition members maximize joint welfare of members max Bi (ei ) − Di (E) X  {ei }i∈C i∈C Non-coalition members maximize own welfare max Bi (ei ) − Di (E) ei We solve game by backward induction R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 98 – The Coalition Formation Game Special Case: only 2 Countries – Stage 2 First, we consider case of only 2 countries Stage 2: 3 possible states of the world: 1 Both countries joined coalition 2 Only one country joined coalition 3 No country joined coalition If both countries join → Global Social Optimum (GSO) B ei = e⋆i = ϵi − , ∀ i = 1,... , n αi In all other cases → Global Emissions Game (GEG) βi ei = êi = ϵi − , ∀ i = 1,... , n αi R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 99 – The Coalition Formation Game Special Case: only 2 Countries – Stage 1 Countries anticipate consequences of their choices in Stage 1: If country i does not join coalition → GEG in stage 2 If country i joins the coalition: → GEG in stage 2 if other country does not join coalition → GSO in stage 2 if other country joins coalition Both countries prefer GSO over GEG if countries sufficiently symmetric or coalition members implement appropriate transfer scheme R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 100 – The Coalition Formation Game Special Case: only 2 Countries – Nash Equilibrium Countries may be indifferent whether to join coalition (if they believe other country does not join coalition) → tie-breaking rule Consider following tie-breaking rule: join coalition if indifferent between joining or not Under this tie-breaking rule both countries join coalition in stage 1 Unique subgame perfect equilibrium (SPE) of Coalition Formation Game: 1 Stage 1: Unconditionally join coalition 2 Stage 2: Play GSO if other player joined coalition, play GEG otherwise R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 101 – The Coalition Formation Game General Case (i) Coalition formation game works well for two players, because players have only 2 choices: 1 Outcome of Global Social Optimum 2 Outcome of Global Emissions Game Obviously, this is different for n > 2 Consider possible states of the world for n = 4: 1 All countries join the coalition → Global Social Optimum 2 Three countries join the coalition 3 Two countries join the coalition 4 One or no country joins the coalition → Global Emissions Game R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 102 – The Coalition Formation Game General Case (ii) Country i’s pay-off Wi depends on whether country i joins coalition and who else is in the coalition C A coalition is stable if ceteris paribus 1 Internal stability: Wi (i ∈ C) > Wi (i ∈ / C) 2 External stability: Wi (i ∈/ C) > Wi (i ∈ C) In particular, countries may have incentive not to join coalition if many other countries join → free-riding R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 103 – The Coalition Formation Game n Symmetric Countries Consider n symmetric countries: αi = α , βi = β , ϵi = ϵ , ∀ i = 1,... , n For symmetric countries only number of coalition members k = |C| matters We solve Coalition Formation Game by backward induction Stage 2: Determine emission choices given k countries joined coalition in stage 1 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 104 – The Coalition Formation Game n Symmetric Countries – Stage 2 (i) Coalition members set emissions to maximize: X  max Bi (ei ) − Di (E) {ei }i∈C i∈C This yields emission levels: Bi′ (ei ) = Dj′ (E) X j∈C kβ α(ϵ − ei ) = kβ ⇒ eC i (k) = ϵ − α R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 105 – The Coalition Formation Game n Symmetric Countries – Stage 2 (ii) Non-Coalition members set emissions to maximize: max Bi (ei ) − Di (E) ei This yields emission levels: Bi′ (ei ) = Di′ (E) β α(ϵ − ei ) = β ⇒ eN C i (k) = ϵ − α Global emissions: β 2 E(k) = keC NC  i (k) + (n − k)ei (k) = E − k + (n − k) α R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 106 – The Coalition Formation Game n Symmetric Countries – Stage 1 (i) Stage 1: Seek k max for which internal and external stability hold First, determine pay-offs depending on coalition size k Coalition members: Wi (i ∈ C, k) = Bi eC

i (k) − Di E(k) 1 β2 h 2   i =ϵ αϵ − nβ + k + 2(n − k) 2 2α Non-coalition members: / C, k) = Bi eN C

Wi (i ∈ i (k) − Di E(k) 1 β2 h 2   i =ϵ αϵ − nβ + 2k − 1 + 2(n − k) 2 2α R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 107 – The Coalition Formation Game n Symmetric Countries – Stage 1 (ii) Second, define stability function Si (k) of country i: Si (k) = Wi (i ∈ C, k) − Wi (i ∈ / C, k − 1) Internal stability requires Wi (i ∈ C, k) − Wi (i ∈ / C, k − 1) > 0 → Si (k) > 0 External stability requires Wi (i ∈ / C, k) − Wi (i ∈ C, k + 1) > 0 → Si (k + 1) < 0 dSi (k) If < 0 there exist unique SPE k max : dk Si (k0 ) = 0 , k max = ⌊k0 ⌋ R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 108 – The Coalition Formation Game n Symmetric Countries – SPE (i) Graphical Illustration: Si (k) kmax k R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 109 – The Coalition Formation Game n Symmetric Countries – SPE (ii) Stability function for n symmetric countries: β2 h 4k − k 2 − 3 i Si (k) = 2α We obtain: dSi (k) β2 = (4 − 2k) ≤ 0 ∀k≥2 dk 2α Si (k0 ) = 0 ⇒ k0 = 3 k max = ⌊k0 ⌋ ⇒ k max = 3 Stable coalition size of Coalition Formation Game is k max = 3 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 110 – The Coalition Formation Game n Symmetric Countries – Summary Stable coalition size k max = 3 independent of exogenous parameters α, β, ϵ and number of countries n! Thus, Coalition Formation Game achieves global social optimum for n≤3 What about n ≥ 4? R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 111 – The Coalition Formation Game n Symmetric Countries – Emissions (i) Consider n ≥ 4 Insert k max = 3 in second stage emission levels: 3β β β eC i (3) = ϵ − , eN C i (3) = ϵ − , E(3) = E − (6 + n) α α α For comparison: n B B GSO: e⋆i = ϵi − E⋆ = E − X , ∀ i = 1,... , n , αi i=1 αi n βi X βi GEG: êi = ϵi − , ∀ i = 1,... , n , Ê = E − αi i=1 αi R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 112 – The Coalition Formation Game n Symmetric Countries – Emissions (ii) Consider α = 0.1n, β = 0.05/n, ϵ = 1/n Graphical illustration: domestic emissions levels êi = eN i C (3) i (3) eC e⋆i R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 113 – The Coalition Formation Game n Symmetric Countries – Emissions (iii) Consider α = 0.1n, β = 0.05/n, ϵ = 1/n Graphical illustration: global emissions levels Ê E(3) E⋆ R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 114 – The Coalition Formation Game n Symmetric Countries – Welfare (i) Insert k max = 3 in welfare levels: 1 β2   Bi e C

Wi (i ∈ C, 3) = − Di E(3) = ϵ i (3) αϵ − nβ + (2n + 3) 2 2α / C, 3) = Bi eN C

Wi (i ∈ i (3) − Di E(3) 1 β2   =ϵ αϵ − nβ + (2n + 11) 2 2α For comparison: 1 β2 2   Wi⋆ Bi e⋆i ⋆

= − Di E = ϵ αϵ − nβ + n 2 2α 1 β2  

Ŵi = Bi êi − Di Ê = ϵ αϵ − nβ + (2n − 1) 2 2α R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 115 – The Coalition Formation Game n Symmetric Countries – Welfare (ii) Consider α = 0.1n, β = 0.05/n, ϵ = 1/n Graphical illustration: domestic social welfare Ŵi WiN C (3) WiC (3) Wi⋆ R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 116 – The Coalition Formation Game n Symmetric Countries – Welfare (iii) Consider α = 0.1n, β = 0.05/n, ϵ = 1/n Graphical illustration: global social welfare Ŵ W (3) W⋆ R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 117 – The Coalition Formation Game Conclusion Coalition Formation Game works well for small number of countries For n ≤ 3 global social optimum can be implemented as SPE Little improvement over GEG for large number of countries CFG predicts international agreements with small number of participating countries → contrary to real world observation R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 118 –  4.2 Modest International Cooperation R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 119 – Modest International Cooperation Motivation Coalition Formation Game predicts small coalition sizes However, real world climate agreements ratified by very large number of countries We slightly generalize CFG to allow for different degrees of ambition with respect to climate change mitigation R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 120 – Modest International Cooperation Model Assumptions (i) Again C denotes set of countries in coalition and k = |C| is number of coalition members In Coalition Formation Game we assumed that coalition members fully internalize emissions externality imposed on all members: X  max Bi (ei ) − Di (E) {ei }i∈C i∈C We now assume that coalition members only account for fraction γ of damages from coalition members X  max Bi (ei ) − γDi (E) {ei }i∈C i∈C R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 121 – Modest International Cooperation Model Assumptions (ii) Again, we consider n identical countries: αi = α , βi = β , ϵi = ϵ , ∀ i = 1,... , n To ensure that behavior for coalition members is incentive compatible we impose: 1/k < γ ≤ 1 If γ < 1/k coalition members abate less than non-coalition members R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 122 – Modest International Cooperation Structure of the Game Modest International Cooperation (MIC) is dynamic (2-stage) game: 1 Countries simultaneously decide whether to join coalition 2 Countries simultaneously decide about emissions: Coalition members maximize: max Bi (ei ) − γDi (E) X  {ei }i∈C i∈C Non-coalition members maximize own welfare max Bi (ei ) − Di (E) ei We solve game by backward induction R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 123 – Modest International Cooperation n Symmetric Countries – Stage 2 (i) Coalition members set emissions to maximize: X  max Bi (ei ) − γDi (E) {ei }i∈C i∈C This yields emission levels: Bi′ (ei ) = γ Dj′ (E) X j∈C kβ α(ϵ − ei ) = γkβ ⇒ eC i (k, γ) = ϵ − γ α R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 124 – Modest International Cooperation n Symmetric Countries – Stage 2 (ii) Non-Coalition members set emissions to maximize: max Bi (ei ) − Di (E) ei This yields emission levels: Bi′ (ei ) = Di′ (e) β α(ϵ − ei ) = β ⇒ eN C i (k, γ) = ϵ − α Global emissions: β 2 E(k, γ) = keC NC  i (k) + (n − k)ei (k) = E − γk + (n − k) α R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 125 – Modest International Cooperation n Symmetric Countries – Stage 1 (i) Stage 1: Seek k max for which internal and external stability holds First, determine pay-offs depending on coalition size k Coalition members: Wi (i ∈ C, k, γ) = Bi eC

i (k, γ) − Di E(k, γ) 1 β2 h 2   i =ϵ αϵ − nβ + γk (2 − γ) + 2(n − k) 2 2α Non-coalition members: / C, k, γ) = Bi eN C

Wi (i ∈ i (k, γ) − Di E(k, γ) 1 β2 h   2γk 2 − 1 + 2(n − k) i =ϵ αϵ − nβ + 2 2α R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 126 – Modest International Cooperation n Symmetric Countries – Stage 1 (ii) Second, derive stability function for n symmetric countries: Si (k, γ) = Wi (i ∈ C, k, γ) − Wi (i ∈ / C, k − 1, γ) β2 h γ(4k − γk 2 − 2) − 1 i = 2α We obtain: ∂Si (k, γ) β2 2 =− [γ(2γk − 4)] ≤ 0 ∀ k ≥ ∂k 2α γ ∂ 2 Si (k, γ) β2γ2 = − < 0 ⇒ Si (k, γ) strictly concave in k ∂k 2 α β2 1 1 Si (2, γ) = [γ(8 − 4γ − 2) − 1] > 0 as γ > = 2α k 2 R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 127 – Modest International Cooperation n Symmetric Countries – Stability Function Consider α = 0.1, β = 0.05, ϵ = 1 Graphical illustration: Stability function Stability function for γ = 1, 0.8, 0.6, 0.4 (from black to light gray) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 128 – Modest International Cooperation n Symmetric Countries – SPE This implies MIC has unique SPE: √ 2+ 3 − 2γ Si (k0 , γ) = 0 ⇒ k0 = γ $ √ % max 2+ 3 − 2γ k = ⌊k0 ⌋ = γ k max is weakly increasing with decreasing γ: 1 √ dk0 γ(3 − 2γ)− 2 + 3 − 2γ =− 0 ∂γ 2α But coalition size k in SPE also depends on γ R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 133 – Modest International Cooperation n Symmetric Countries – Welfare (ii) Consider α = 1, β = 0.05, ϵ = 1, n = 10 Graphical illustration: k max in SPE Stable coalition size kmax as a function of γ R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 134 – Modest International Cooperation n Symmetric Countries – Welfare (iii) Consider α = 1, β = 0.05, ϵ = 1, n = 10 Graphical illustration: Global social welfare in SPE Global social welfare in SPE as a function of γ (red: global welfare in GEG, green: global welfare in GSO) R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 135 – Modest International Cooperation Conclusion Modest International Cooperation may implement any coalition size (including grand coalition) as SPE However, larger coalition sizes come at cost of lower γ Larger but less ambitious coalitions (may) achieve more (in terms of global social welfare) than small and ambitious coalition MIC may perform better than CFG but falls short of GSO for n > 3 In addition: CFG and MIC do not explain how cooperation within coalition is achieved R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 136 –  4.3 Literature R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 137 – Literature for Section 4 U. Wagner (2001): The design of stable international environmental agreements: economic theory and political economy. Journal of Economic Surveys (15): 377–411. M. Finus & S. Maus (2008): Modesty may pay! Journal of Public Economic Theory (10): 801–826. R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 138 –  5 International Permit Markets R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 139 – International Permit Markets Motivation (i) We have seen that CFG and MIC fall short of GSO For CFG: small coalition sizes and not much improvement over GEG for large number of countries n For MIC : modest but large agreements may achieve more then GEG But results rest on two crucial assumptions: 1 symmetric countries 2 cooperation within coalition by assumption We now explore bilateral forms of international cooperation R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 140 – International Permit Markets Reminder: Global Emissions Game Recall outcome of Global Emissions Game: βi X βi êi = ϵi − , Ê = ϵ − αi i αi Inefficiencies of Global Emissions Game (GEG): 1 Maximizing domestic welfare countries do not take into account negative externalities emissions impose on other countries → global emissions too high 2 Maximizing domestic welfare does, in general, not lead to equal marginal benefits across countries → wrong distribution of global emissions R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 141 – International Permit Markets Motivation (ii) Marginal benefits in NE of GEG: Bi′ (ei ) = βi ̸= βj = Bj′ (ej ) Even if we cannot get global emissions right can we at least provide cost-effective emission reductions? Yes → International Emission Permit Market: In market equilibrium → unique permit price p Maximizing domestic welfare countries buy or sell permits such that Bi′ (ei ) = p R. Winkler (VWI & OCCR) Climate Economics: Intl. Cooperation Fall Term 2024 – 142 –