Rational Choice Models and International Conflict

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Questions and Answers

Explain briefly the meaning of the following game-theoretical concepts:
(a) Decision node

A point in the game tree where a player must make a decision.

Explain briefly the meaning of the following game-theoretical concepts:
(b) Best Response

A strategy that is strictly worse than another strategy in one contingency and not better in any contingency.

Explain briefly the meaning of the following game-theoretical concepts:
(c) Weakly dominated strategy

A piece of information is common knowledge if both (all) players know it, know that all players know it, know that all players know that all players know it, and so on ad infinitum.

Explain briefly the meaning of the following game-theoretical concepts:
(d) Common knowledge

<p>An information set conveys the information a player has about previous moves in the game when the player concerned must make a decision.</p> Signup and view all the answers

Explain briefly the meaning of the following game-theoretical concepts:
(e) Information set

<p>The players' strategic types (strategy sets and preferences) are not known by at least one player</p> Signup and view all the answers

Explain briefly the meaning of the following game-theoretical concepts:
(f) Incomplete information

<p>The history of the game is always common knowledge.</p> Signup and view all the answers

Explain briefly the meaning of the following game-theoretical concepts:
(g) Perfect information

<p>A subgame starts with a singleton, encompasses all subsequent decision nodes, and does not cut across any information set.</p> Signup and view all the answers

Explain briefly the meaning of the following game-theoretical concepts:
(h) Pareto optimal

<p>A set of strategies that are best responses to each other for every subgame, given the players' beliefs. 2. The players' beliefs are updated along the equilibrium path, using Bayes' rule (wherever possible).</p> Signup and view all the answers

Explain briefly the meaning of the following game-theoretical concepts:
(i) Subgame

<p>Two definitional criteria: 1. A set of strategies that are best responses to each other for every subgame, given the players' beliefs. 2 The players' beliefs are updated along the equilibrium path, using Bayes' rule (wherever possible).</p> Signup and view all the answers

Explain briefly the meaning of the following game-theoretical concepts:
(j) Bayesian perfect equilibrium

<p>A point in the game tree where a player must make a decision.</p> Signup and view all the answers

What is p2 in terms of c1 and c2?

<p>C2/(C1+C2)</p> Signup and view all the answers

Find the subgame-perfect equilibrium of the game when it is known that the military capability of State 2 is three times that of State 1, c2 = 3c1.

<p>SPNE = {(Yield, War)}</p> Signup and view all the answers

Find the subgame-perfect equilibrium of the game when it is common knowledge that the military capabilities of state 1 and state 2 are equal, c1 = c2.

<p>SPNE = {(SF, War)}</p> Signup and view all the answers

Based on this model, how does the balance of power effect the likelihood of war? Please explain your answer.

<p>Increased balance of power increases the likelihood of conflict.</p> Signup and view all the answers

Find the subgame-perfect equilibrium of the game both when the cost is high c = -15 and when the cost is low c = -5.

<p>When the cost is low (c = -5), SPNE = {DVN, HS} and when the cost is high (c = -15), SPNE = {VN, DN}.</p> Signup and view all the answers

Show the game in normal form when the cost is high c = -15.

<h2>Country 2 Country 1 | DVN | VN | DN</h2> <p>DVN | 0, 0 | 40, -10 | 0, 0 VN | -10, -15 | 0, 0 | -10, -15 DN | 0, 0 |40, -10 | 0, 0</p> Signup and view all the answers

When the cost is high (c = -15), does the game have any Nash equilibria other than the subgame-perfect equilibrium?

<p>True (A)</p> Signup and view all the answers

When the cost is high (c = -15), if there are any Nash equilibria besides the subgame-perfect equilibrium, list them.

<p>NE = {(VN, DN), (DVN, HS)} where SPNE = {VN,DN}.</p> Signup and view all the answers

When the cost is high (c = -15), if there are any Nash equilibria besides the subgame-perfect equilibrium, explain in words why this outcome is not credible. Would such a Nash equilibrium be an "effective threat"?

<p>The NE (VN, DN) is not credible because Country 2 would not rationally do nothing. They would deviate, given the order of play. This threat would not be an &quot;effective threat&quot; because Country 1 is not going to choose not to violate the norm because of the threat of sanctions.</p> Signup and view all the answers

Redraw the game tree to account for the uncertainty of Country 1.

<p>Country 1 DVN<br /> VN<br /> Country 2 DN<br /> HS<br /> p Nature 1-p Country 1 DVN<br /> VN<br /> Country 2 DN<br /> HS<br /> 0, 0 0, 0 40, -10 -10, -15 40, -10 -10, -5</p> Signup and view all the answers

Find the perfect Bayesian equilibria for possible values of p.

<p>PBE = {(DVN, (DN, HS);p &lt; 1/5), (VN, (DN, HS); p ≥ 1/5)}</p> Signup and view all the answers

Are they separating or pooling equilibria?

<p>True (A)</p> Signup and view all the answers

For at least one equilibrium, use Bayes' rule to find the posterior belief of Country 1 at the end of the game about whether the cost of sanctioning was high c = -15 given the actions of Country 2.

<p>Pr(High¦DN) = p / (p + (1-p)×0) = 1, Pr(High¦HS) = 0 / (p + (1-p)×1) = 0</p> Signup and view all the answers

Given a known probability of winning the war, p = 0.45, and the corresponding values of p - CUSA and p+CUSSR, constituting the respective costs of war for the USA and the USSR. Would you expect war under these conditions? Why?

<p>Under conditions of complete information, war is unlikely.</p> Signup and view all the answers

Now consider a situation, referring to Figure, in which the costs of war for the Soviet Union are uncertain. They are either p + CUSSR or p + CUSSR. If the USA does not know the costs of war for the USSR, what are the implications? Would you expect war? Why?

<p>War is possible. The US will be tempted to make as big a demand as possible, but this could lead to conflict if the USSR has low costs of war. The USA faces a dilemma, not knowing the USSR's true intentions, and the potential risk of escalation.</p> Signup and view all the answers

Set pCUSA = 0.35, p + CUSSR = 0.55, and p + CUSSR = 0.65. Q is the probability that the USSR has high costs of war, p+CUSSR = 0.55. Given this game of one-sided incomplete information, what are the subgame perfect equilibria?

<p>The subgame perfect equilibria is the US making the larger offer when (2/3 &lt; q); When (2/3 &gt; q), the US should make the offer that both types will accept (.55). SPE = {(high demand, (war, peace bargain); q &gt; 2/3} and {(low demand, (war, peace bargain); q &lt; 2/3}.</p> Signup and view all the answers

During the height of the Cold War, both the USSR and USA had the capability to destroy the other side after a first-strike attack. This was referred to as Mutual Assurred Destruction (MAD). Referring to Figure, what happens to the costs of war for both sides in this situation? Would you expect war? Why?

<p>MAD creates a scenario where the costs of war for both sides are extremely high. Given the capability for mutual destruction, engaging in conflict is highly irrational and likely to lead to unacceptable consequences. Under MAD, war becomes a very unlikely prospect, as the potential for destruction outweighs any potential gains.</p> Signup and view all the answers

Flashcards

Decision Node

A point in the game tree where a player has to make a decision.

Best Response

The best strategy a player can choose given the other players' strategies.

Weakly Dominated Strategy

A strategy that is always strictly worse than another strategy, regardless of what other players do.

Common Knowledge

Information that is known to all players, and they know that others know it, and so on.

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Information Set

A set of decision nodes where a player has the same information about previous moves in the game.

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Incomplete Information

When players do not know the other players' strategies, preferences, or payoffs.

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Perfect Information

When the history of the game is always known to all players.

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Pareto Optimal

An outcome where no player can improve their payoff without making another player's payoff worse.

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Subgame

A part of the game tree that starts at a single decision node and includes all subsequent decisions.

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Bayesian Perfect Equilibrium

A set of strategies that are best responses to each other in every subgame, considering players' beliefs.

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p2

The probability of State 2 winning the war.

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c1

The military capability of State 1.

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c2

The military capability of State 2.

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Subgame Perfect Equilibrium (SPNE)

A solution to the game that is optimal for both players in every subgame.

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Harsh Sanctions (HS)

A strategy for Country 2 where they impose harsh sanctions if Country 1 violates the norm.

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Do Nothing (DN)

A strategy for Country 2 where they do nothing if Country 1 violates the norm.

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Violate Norm (VN)

A strategy for Country 1 where they choose to violate the international norm.

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Don't Violate Norm (DVN)

A strategy for Country 1 where they choose not to violate the international norm.

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c

The cost to Country 2 of imposing sanctions.

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Not Credible

A strategy that leads to an outcome that is not credible, meaning one player would not rationally choose it.

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p

The probability that Country 2 faces a high cost of imposing sanctions.

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Bayes' Rule

A framework for updating a belief about an event based on new evidence.

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Separating Equilibrium

An equilibrium where different types of players take different actions.

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Pooling Equilibrium

An equilibrium where all types of players take the same action.

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p

The probability of winning a war.

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CU SA

The cost of war for the USA.

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CU SSR

The cost of war for the USSR.

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Mutual Assured Destruction (MAD)

An situation where both sides have the ability to destroy each other in a nuclear war.

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One-Sided Incomplete Information

A situation where one player has uncertainty about another player's information about the game.

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q

The probability of the USSR facing high costs of war.

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Study Notes

EXAM STV4217B

  • Rational Choice Models and International Conflict
  • Autumn 2023
  • Date: 20 December 2023
  • Time: 15:00
  • Duration: 3 hours

Question 1 (20%)

  • Decision node: A point in a game tree where a player must make a decision.
  • Best Response: A strategy that yields the highest payoff for a player given opponent's choice.
  • Weakly Dominated Strategy: A strategy that is never better than another, and in one or more cases, strictly worse.
  • Common Knowledge: The information that all players know, and they know that everyone else knows it, and so on.
  • Information Set: Information a player has about prior moves when the player makes a decision.
  • Incomplete Information: Players' strategies and preferences are unknown to at least one player.
  • Perfect Information: The history of the game is always common knowledge.
  • Pareto Optimal: The strategy that can't be further improved without harming another player.
  • Subgame: A subgame starts from a single point and covers all subsequent decision points without crossing any information sets.
  • Bayesian Perfect Equilibrium: Strategies that are best responses to each other, considering players' beliefs that are updated along the equilibrium path using Bayes' rule.

Question 2 (25%)

  • Threat of War: State 2 threatens State 1 with war unless State 1 closes suspected WMD facilities.
  • Probability of Victory (p1): Probability of State 1 winning the war is 0.125; the probability of State 2 winning is 1-p1.
  • Military Capability (C1, C2): C2 is 3 times C1.
  • Subgame Perfect Equilibrium (SPNE): The equilibrium of a game when both players make the best choices to reach a stable outcome.
  • Influence of Military Capabilities on War Likelihood: A balance of power influences the likelihood of war.
  • Impact of Common Knowledge of Military Capabilities on War: Equal military capability between states reduces the likelihood of war when common knowledge is established.

Question 3 (25%)

  • International Norm Violation: Country 1 can violate an international norm, Country 2 can impose sanctions.
  • Cost of Sanctions (c): Sanctions have either a high or low cost for Country 2 (-15 or -5).
  • Subgame Perfect Equilibrium (SPNE): The best possible outcome in a game, considering all possible future moves.
  • Nash Equilibrium (NE): A solution that occurs when no player has incentive to change their strategy, given other players' choice.
  • Credibility of Threats: A credible threat in game theory is one where threats are believable and will be carried out, even in the case of incomplete information.
  • Incompletely Known Costs: Country 1 does not know the cost, which can be high or low.
  • Bayesian Nash Equilibrium: Players adjust their behaviour based upon a probability distribution of outcomes.

Question 4 (25%)

  • Cold War Context: The question examines the strategic choices during the Cold War.
  • Probability of Winning War (p): The probability of winning a war is explicitly given as 0.45.
  • Costs of War (CUSA, CUSSR): Costs of war for US and Soviet Union are mentioned but not quantified.
  • Mutual Assured Destruction (MAD): A concept in which the high costs of war make a first strike unlikely.
  • Uncertain Costs of War: Costs of war for one of the sides are uncertain.
  • Incomplete Information: The other side does not know whether the other side has high costs of war, meaning a level of uncertainty exists.
  • Subgame Perfect Equilibria (SPNE): The optimal outcomes, considering all possible future strategic choices.

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