Economic Game Theory Tutorial: Solve Econ 0200 Final Exam Problems

Introduction to Economic Game Theory

Economic game theory is the study of strategic decision-making. Whether you're analyzing a prisoner's dilemma or a coordination game, the concepts are fundamental to economics, business, and even everyday life. In this tutorial, we'll walk through the types of problems you might encounter on an exam like Econ 0200, using the provided game matrices as a guide. We'll cover Pareto optimality, Nash equilibrium (NE), evolutionary stable strategies (ESS), repeated games, and present value calculations.

Pareto Optimality: Finding the Best Outcomes

An outcome is Pareto optimal if no player can be made better off without making another player worse off. For Game 1 (A: 1,1 / B: 4,15 / C: 15,4 / D: 7,7), the Pareto optimal outcomes are (B,A) and (A,B) because (A,A) is dominated by both. In Game 2 (12,12 / 5,12 / 12,5 / 16,16), all outcomes except (A,A) are Pareto optimal? Actually, (A,A) gives 12 each, but (B,B) gives 16 each, so (A,A) is not Pareto optimal. (A,B) and (B,A) give 5 and 12, so they are also Pareto optimal? Wait, (B,B) gives 16,16 which is better for both than (A,B) or (B,A), so those are not Pareto optimal. Only (B,B) is Pareto optimal. Similarly, check each game.

Nash Equilibrium in Pure Strategies

A Nash equilibrium occurs when each player's strategy is a best response to the other's. For Game 1: Check each cell. (A,A): Player 1 gets 1, but if they switch to B they get 4, so not NE. (A,B): Player 1 gets 15, switching to B gives 7, so no incentive; Player 2 gets 4, switching to A gives 15, so incentive to switch? Actually, Player 2 gets 4 from (A,B), if they switch to A they get 1, so no incentive? Wait, careful: In (A,B), Player 1 plays A, Player 2 plays B. Player 1's payoff: 15. If Player 1 switches to B, payoff becomes 7 (since (B,B) gives 7). So no incentive. Player 2's payoff: 4. If Player 2 switches to A, payoff becomes 1 (since (A,A) gives 1). So no incentive. So (A,B) is a Nash equilibrium. Similarly, (B,A) is also NE. (B,B): Player 1 gets 7, switching to A gives 15, so incentive, not NE. So pure NE: (A,B) and (B,A).

Named Games: Identifying the Classic

Game 1 has payoffs that resemble a chicken game? Actually, chicken has (0,0) for both swerve? Here, the best outcome for each is to play different strategies: (15,4) and (4,15) are the best, while (1,1) is worst. That fits the definition of a coordination game? No, coordination games have mutual best outcomes like (1,1) and (2,2). This is actually a game of chicken (also called hawk-dove). So for Game 1, name: chicken. For Game 2, the payoffs are (12,12) for (A,A), (5,12) for (A,B), (12,5) for (B,A), and (16,16) for (B,B). This is a prisoner's dilemma? In PD, defection dominates cooperation and mutual defection is worse than mutual cooperation. Here, (B,B) gives 16 > 12, and (A,B) gives 5 for player 1? Actually, check: If both cooperate (A,A): 12 each. If one defects (B) and other cooperates (A), defector gets 12? Wait, (B,A) gives player 1 (B) 12, player 2 (A) 5? No, the matrix is: row A: (12,12) and (5,12); row B: (12,5) and (16,16). So if player 1 plays B and player 2 plays A, player 1 gets 12, player 2 gets 5. So defecting when other cooperates gives 12 vs 5 for cooperating? That's not typical PD where defector gets a higher payoff. Actually, this is a stag hunt? Stag hunt has (4,4) for both hunt stag, (3,3) for both hare, and (0,0) for mismatched? Not exactly. This game is actually a coordination game with a payoff-dominant equilibrium (B,B) and a risk-dominant? But the problem asks to name from seven named games: prisoner's dilemma, chicken, stag hunt, battle of the sexes, etc. Given the payoffs, (B,B) is best for both, and (A,A) is second best, so it's a coordination game (like stag hunt). So name: stag hunt.

Evolutionary Stable Strategies (ESS)

An ESS is a strategy that, if adopted by a population, cannot be invaded by a mutant strategy. For Game 1, test A: Assume all play A. Fitness of A = 1 (when playing against A). Fitness of B = 4 (when playing against A, since B vs A gives 4). Since B's fitness (4) > A's fitness (1), A is not ESS. For B: all play B, fitness of B = 7 (B vs B), fitness of A = 15 (A vs B). Since A's fitness (15) > B's (7), B is not ESS. So no pure ESS. Check mixed? The static midpoint in evolutionary setting: if a mixed strategy exists where both have equal fitness. Let p be proportion of A. Fitness of A = p*1 + (1-p)*15 = 15 - 14p. Fitness of B = p*4 + (1-p)*7 = 7 - 3p. Set equal: 15 - 14p = 7 - 3p => 8 = 11p => p = 8/11. So static midpoint exists at p=8/11. Is it stable? Check derivative: d(fitA - fitB)/dp = -14 - (-3) = -11 < 0, so stable. So for Game 1, static midpoint exists and is stable.

Present Value Calculations

Present value (PV) = Future Value / (1 + r)^n. For example, $1,000,000 in 9 years at 7%: PV = 1,000,000 / (1.07)^9 = 1,000,000 / 1.838459 = about $543,934. Similarly for others. In today's high-interest environment (2026, rates are around 5-7%), these calculations are crucial for investment decisions.

Repeated Games and Cooperation

In infinitely repeated games, cooperation can be sustained if the discount factor (or interest rate) is sufficiently high. For Game 4 (7,7 / 26,6 / 6,26 / 19,19), this is a prisoner's dilemma? Check: mutual cooperation (A,A) gives 7 each, but defecting (B) when other cooperates gives 26? Actually, (A,B): player 1 gets 26, player 2 gets 6. So defection payoff is 26, temptation. Mutual defection (B,B) gives 19. So it's a prisoner's dilemma if 19 < 7? No, 19 > 7, so mutual defection is better than mutual cooperation? That's not PD. Actually, this is a chicken game? (A,A) gives 7, (B,B) gives 19, but (A,B) gives 26 for row, 6 for column. So the best outcome is to be the defector when other cooperates. This is actually a deadlock? But anyway, to sustain cooperation (A,A) in infinitely repeated game with grim trigger, the condition is: cooperate if PV(cooperate) >= PV(deviate). For grim trigger, cooperate forever gives 7 each period. Deviate gives 26 in first period, then 19 forever after. So condition: 7/(1 - δ) >= 26 + δ*19/(1 - δ). Solve for δ: 7/(1-δ) >= 26 + 19δ/(1-δ) => 7 >= 26(1-δ) + 19δ => 7 >= 26 - 26δ + 19δ => 7 >= 26 -7δ => 7δ >= 19 => δ >= 19/7 ≈ 2.714, impossible. So no interest rate can sustain cooperation. For tit-for-tat, similar condition yields no cooperation. So for Game 4, cooperation not sustainable.

Best Response Graphs and Evolutionary Graphs

Drawing best response graphs involves plotting each player's best response to the other's strategy. For a 2x2 game, you can plot the probability of playing A for each player. The intersection points are Nash equilibria. Evolutionary graphs show how the proportion of A changes over time. For Game 1, the evolutionary graph would show an unstable equilibrium at p=0 and p=1, and a stable interior equilibrium at p=8/11.

Application to Real-World Trends

Game theory is everywhere. In 2026, AI companies like OpenAI and Google are in a prisoner's dilemma over AI safety regulations. Each can choose to cooperate (implement safety checks) or defect (rush to market). The payoff matrix often resembles a prisoner's dilemma where mutual cooperation yields long-term benefits, but defection gives short-term gains. Similarly, in sports, teams choose strategies (offense/defense) that can be modeled as a coordination game. Understanding Nash equilibrium helps predict outcomes.

Conclusion

By mastering these concepts—Pareto optimality, Nash equilibrium, ESS, present value, and repeated games—you can tackle any game theory exam. Practice with the provided games and check your answers. Remember, the key is to systematically evaluate each cell and use the definitions.