Lecture 1 Introduction
This lecture reviews 2 basic tools necessary for this course, Game Theory and Political
Economy Basics.
Game theory
2 types of games:
1. Normal form (simultaneous) game
Nash equilibrium: A Nash equilibrium (NE) is a pair of strategies, one for each player, such
that no player has an incentive to deviate from their strategy given the strategy of the other
player. Stated differently, each player’s NE strategy is a best response to the other player’s
strategy
➔ Few examples
2. Extensive form (sequential) game
Subgame perfect equilibrium: A subgame perfect equilibrium (SPE) is a pair of strategies, one
for each player, which is a Nash equilibrium at each decision node.
We use backwards induction: start at the end of the game
➔ Few examples
Strategic dominance
Dominated
A strategy a ∈ XA (b ∈ XB) is weakly dominated if there is another strategy a′ ∈ XA (b′ ∈ XB)
that yields a weakly higher payoff to player A (player B) for all possible strategies of player B
(player A) — i.e. πA(a ′ , b) ≥ πA(a, b) for all b ∈ XB (πB(a, b ′ ) ≥ πB(a, b) for all a ∈ XA). If the
inequality is strict, we say that strategy a (b) is strictly dominated
Dominant
A strategy a ∈ XA (b ∈ XB) is weakly dominant if it yields weakly higher payoffs than any
other strategy a′ ∈ XA to player A (b′ ∈ XB to player B) for all possible strategies of player B
(player A) — i.e. for any b ∈ XB (a ∈ XA),πA(a, b) ≥ πA(a ′ , b) for all a′ ∈ XA (πB(a, b) ≥ πB(a, b
′ ) for all b′ ∈ XB). If the inequality is strict, we say that strategy a (b) is strictly dominant.
Iterated deletion of dominated strategies: we can delete dominated strategies
Games with a continuum of actions: player chooses an action from a continuous action set
on the real line. Payoffs are determined by a payoff function (continuous + differentiable).
Simultaneous moves: derive players best-response functions by maximizing their payoffs
(derivative with respect to action = 0 ). Then solve system of equations.
Sequential move: derive best-response function of the last player by maximizing its payoff,
then plug this into the payoff function of the first player and maximize his payoff too. Then
derive the optimal action of the last player following the first player.
➔ Example (not very hard)
,Political Economy Basics
Two forms of democracy:
1. Direct democracy
2. Indirect or representative democracy
A voting problem structure:
1. Voters: N = {1, 2, …, n}
2. Alternatives: A = {a1, a2, …, ak}
3. Preferences: are given by indirect utility function Vi(a)
4. Voting rule
Several voting rules:
1. Majority rule: holding sequential pairwise majority comparisons.
2. Plurality rule: voters choose their favourite alternative from the complete set of
alternatives and the alternative with the most votes wins
3. Plurality rule with runoff: if no alternative receives a majority in the first round, the
two alternatives with the most votes face off in the second round in a runoff election
4. Borda-count: score-based voting rule where each voter ranks all k alternatives.
Alternative with the highest total scare wins.
With two alternatives, al four voting rules are equivalent.
For three or more alternatives → never an ideal voting rule:
Arrow’s impossibility theorem
If there are three or more alternatives, there does not exist a voting rule that satisfies the
following properties:
1. Universal domain: All preferences are allowed.
2. Unanimity: If all voters agree on what the best alternative is, then that alternative is
chosen.
3. Non-dictatorship: There is no voter such that his or her preferences always
determine which alternative is chosen, regardless of everybody else’s preferences.
4. Independence of irrelevant alternatives: The aggregate ranking between any pair of
alternatives only depends on the individual voters’ rankings between those two
alternatives and is independent of the voters’ preferences over other (irrelevant)
alternatives.
Example of majority rule shows that depending on the order of voting, all alternatives may
win. The agenda-setter can determine the election outcome.
Condorcet:
Condorcet winner: A Condorcet winner (CW), if it exists, is an alternative that defeats all
other alternatives in pair-wise majority comparisons. If a CW exists, it is, by definition,
unique.
Condorcet loser: A Condorcet loser (CL), if it exists, is an alternative that is defeated by all
other alternatives in pair-wise majority comparisons. If a CL exists, it is, by definition, unique.
Condorcet consistency: A voting rule is Condorcet consistent (CC) if it always selects the CW
whenever it exists.
, Trade-off: CC vs. power of agenda setter when CW does not exist
Another example shows:
- Plurality rule can select a CL, hence it is not CC. But plurality rule is very simple/easy.
- Plurality with run off rule never selects a CL, since it would lose in the last stage. But
it is not CC since CW is not selected.
- Borda count never selects a CL, but it might not select CW and hence is not CC
Borde count has an additional undesirable feature, namely that taking away a choice that
was dominated lead to a complete reversal in the ordering of group preferences. → Texas
council example.
Median Voter Theorem
Single-peaked preferences: An indirect utility function V(a) over the set of alternatives A = [a,
a] ⊂ R is single-peaked if it has a unique maximum over the support A. Mathematically, this
means that one of the following conditions is met:
1. V ′ (a) = 0 has a unique solution a∗ in A and V′′(a) < 0 on the entire support A. In this
case, a∗ is the function’s (single) peak.
2. V ′ (a) > 0 on the entire support A, in which case the upper bound of A, a, is the
function’s (single) peak
3. V′ (a) < 0 on the entire support A, in which case the lower bound of A, a, is the
function’s (single) peak.
Median voter theorem I (MVT I): If the space of alternatives is uni-dimensional and ordered
and the preferences for all voters are single-peaked, then:
1. A Condorcet winner (CW) always exists.
2. The CW is the median of the distribution of peaks in the population of the voters.
➔ Sufficient (not necessary) conditions
Discussion
Ranking only valuable if preferences are single-peaked with respect to that characteristic
used to make the ranking.
Example with 300 000 voters who choose level p
Gibbard-Satterthwaite theorem: if voters are voting over three or more alternatives and the
voting rule is non-dictatorial, then the voting procedure is open to manipulation.
Lecture 2 Voters
This lecture considers the question why someone votes, considering 3 models.
Downsian: vote if pei Bei + Dei ≥ C ei
Where p: probability pivotal vote
B: indirect benefit of inducing desired election outcome
D: direct benefit of voting in election
C: cost of voting
The 3 models all endogenize P, D or C
This lecture reviews 2 basic tools necessary for this course, Game Theory and Political
Economy Basics.
Game theory
2 types of games:
1. Normal form (simultaneous) game
Nash equilibrium: A Nash equilibrium (NE) is a pair of strategies, one for each player, such
that no player has an incentive to deviate from their strategy given the strategy of the other
player. Stated differently, each player’s NE strategy is a best response to the other player’s
strategy
➔ Few examples
2. Extensive form (sequential) game
Subgame perfect equilibrium: A subgame perfect equilibrium (SPE) is a pair of strategies, one
for each player, which is a Nash equilibrium at each decision node.
We use backwards induction: start at the end of the game
➔ Few examples
Strategic dominance
Dominated
A strategy a ∈ XA (b ∈ XB) is weakly dominated if there is another strategy a′ ∈ XA (b′ ∈ XB)
that yields a weakly higher payoff to player A (player B) for all possible strategies of player B
(player A) — i.e. πA(a ′ , b) ≥ πA(a, b) for all b ∈ XB (πB(a, b ′ ) ≥ πB(a, b) for all a ∈ XA). If the
inequality is strict, we say that strategy a (b) is strictly dominated
Dominant
A strategy a ∈ XA (b ∈ XB) is weakly dominant if it yields weakly higher payoffs than any
other strategy a′ ∈ XA to player A (b′ ∈ XB to player B) for all possible strategies of player B
(player A) — i.e. for any b ∈ XB (a ∈ XA),πA(a, b) ≥ πA(a ′ , b) for all a′ ∈ XA (πB(a, b) ≥ πB(a, b
′ ) for all b′ ∈ XB). If the inequality is strict, we say that strategy a (b) is strictly dominant.
Iterated deletion of dominated strategies: we can delete dominated strategies
Games with a continuum of actions: player chooses an action from a continuous action set
on the real line. Payoffs are determined by a payoff function (continuous + differentiable).
Simultaneous moves: derive players best-response functions by maximizing their payoffs
(derivative with respect to action = 0 ). Then solve system of equations.
Sequential move: derive best-response function of the last player by maximizing its payoff,
then plug this into the payoff function of the first player and maximize his payoff too. Then
derive the optimal action of the last player following the first player.
➔ Example (not very hard)
,Political Economy Basics
Two forms of democracy:
1. Direct democracy
2. Indirect or representative democracy
A voting problem structure:
1. Voters: N = {1, 2, …, n}
2. Alternatives: A = {a1, a2, …, ak}
3. Preferences: are given by indirect utility function Vi(a)
4. Voting rule
Several voting rules:
1. Majority rule: holding sequential pairwise majority comparisons.
2. Plurality rule: voters choose their favourite alternative from the complete set of
alternatives and the alternative with the most votes wins
3. Plurality rule with runoff: if no alternative receives a majority in the first round, the
two alternatives with the most votes face off in the second round in a runoff election
4. Borda-count: score-based voting rule where each voter ranks all k alternatives.
Alternative with the highest total scare wins.
With two alternatives, al four voting rules are equivalent.
For three or more alternatives → never an ideal voting rule:
Arrow’s impossibility theorem
If there are three or more alternatives, there does not exist a voting rule that satisfies the
following properties:
1. Universal domain: All preferences are allowed.
2. Unanimity: If all voters agree on what the best alternative is, then that alternative is
chosen.
3. Non-dictatorship: There is no voter such that his or her preferences always
determine which alternative is chosen, regardless of everybody else’s preferences.
4. Independence of irrelevant alternatives: The aggregate ranking between any pair of
alternatives only depends on the individual voters’ rankings between those two
alternatives and is independent of the voters’ preferences over other (irrelevant)
alternatives.
Example of majority rule shows that depending on the order of voting, all alternatives may
win. The agenda-setter can determine the election outcome.
Condorcet:
Condorcet winner: A Condorcet winner (CW), if it exists, is an alternative that defeats all
other alternatives in pair-wise majority comparisons. If a CW exists, it is, by definition,
unique.
Condorcet loser: A Condorcet loser (CL), if it exists, is an alternative that is defeated by all
other alternatives in pair-wise majority comparisons. If a CL exists, it is, by definition, unique.
Condorcet consistency: A voting rule is Condorcet consistent (CC) if it always selects the CW
whenever it exists.
, Trade-off: CC vs. power of agenda setter when CW does not exist
Another example shows:
- Plurality rule can select a CL, hence it is not CC. But plurality rule is very simple/easy.
- Plurality with run off rule never selects a CL, since it would lose in the last stage. But
it is not CC since CW is not selected.
- Borda count never selects a CL, but it might not select CW and hence is not CC
Borde count has an additional undesirable feature, namely that taking away a choice that
was dominated lead to a complete reversal in the ordering of group preferences. → Texas
council example.
Median Voter Theorem
Single-peaked preferences: An indirect utility function V(a) over the set of alternatives A = [a,
a] ⊂ R is single-peaked if it has a unique maximum over the support A. Mathematically, this
means that one of the following conditions is met:
1. V ′ (a) = 0 has a unique solution a∗ in A and V′′(a) < 0 on the entire support A. In this
case, a∗ is the function’s (single) peak.
2. V ′ (a) > 0 on the entire support A, in which case the upper bound of A, a, is the
function’s (single) peak
3. V′ (a) < 0 on the entire support A, in which case the lower bound of A, a, is the
function’s (single) peak.
Median voter theorem I (MVT I): If the space of alternatives is uni-dimensional and ordered
and the preferences for all voters are single-peaked, then:
1. A Condorcet winner (CW) always exists.
2. The CW is the median of the distribution of peaks in the population of the voters.
➔ Sufficient (not necessary) conditions
Discussion
Ranking only valuable if preferences are single-peaked with respect to that characteristic
used to make the ranking.
Example with 300 000 voters who choose level p
Gibbard-Satterthwaite theorem: if voters are voting over three or more alternatives and the
voting rule is non-dictatorial, then the voting procedure is open to manipulation.
Lecture 2 Voters
This lecture considers the question why someone votes, considering 3 models.
Downsian: vote if pei Bei + Dei ≥ C ei
Where p: probability pivotal vote
B: indirect benefit of inducing desired election outcome
D: direct benefit of voting in election
C: cost of voting
The 3 models all endogenize P, D or C
