Strategic Behaviour

1
Strategic Behaviour
2
Strategic Voting

• Voter wants to vote NDP
• Risks a wasted vote
• Votes Liberal to prevent Conservative win
• Voting for 2nd favourite to defeat worst outcome

3
Sophisticated Voting

• A stylized example
• 3-person legislature i, j, k
• Voting on pay-raise for themselves
• All really want raise
• Fear voters anger
• Each legislator prefers other two vote Yea

4
Legislators Preference Ordering

• Vote No, bill passes (3)
• Vote Yes, bill passes (2)
• Vote No, bill fails (1)
• Vote Yea, bill fails (0)

5
Modelling the Voting Procedure
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
6
How should i, j k vote?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
7
What does i know about k?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
8
What does i know about k?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
9
What does i know about k?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
10
What does i know about k?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
11
How does i think j will react to k?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
12
How does i think j will react to k?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
13
How does i think j will react to k?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
14
Given that, how should i vote?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
15
Given that, how should i vote?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
16
Given that, how should i vote?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
17
Equilibrium

• Nash equilibrium
• Backward Induction
• Subgame perfect
• No non-credible threats allowed

18
Why cant j threaten to vote No?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
19
It is not credible for j to vote No?
k
Y
2, 2, 2
Y
j
2, 2, 3
N
Y
Y
2, 3, 2
k
N
N
0, 1, 1
i
Y
k
3, 2, 2
N
Y
j
N
1, 0, 1
Y
k
1, 1, 0
N
1, 1, 1
N
20
How might j make threat credible?

• Post a bond
• Give speech vs. raises
• Reputational cost
• Delegate
• Take poll of constituents
• Give a proxy vote to speaker

21
How might j make threat credible?

• Post a bond
• Give speech vs. raises
• Reputational cost
• Delegate
• Take poll of constituents
• Give a proxy vote to speaker

22
How might j make threat credible?

• Delay
• Not show up force i to vote Yea
• Logroll
• Link vote to something i wants
• Change voting procedure
• Vote on voices
• Secret ballot (?)

23
How might j make threat credible?

• Delay
• Not show up force i to vote Yea
• Logroll
• Link vote to something i wants
• Change voting procedure
• Vote on voices
• Secret ballot (?)

24
How might j make threat credible?

• Delay
• Not show up force i to vote Yea
• Logroll
• Link vote to something i wants
• Change voting procedure
• Vote on voices
• Secret ballot (?)

25
Effect of secret ballot

• Three legislators, i, j, k voting on pay raise
  for themselves
• Want the raise, but want to avoid public anger
• Outcome i votes No, forces j k to vote Yes
  face the voters

26
Effect of secret ballot

• Say the vote had been by secret ballot, i, j, k
  voting simultaneously
• What outcome would we see?

27
Outcomes under Secret Ballot

• Under secret ballot, only overall result is
  visible, so 4 possible outcomes
• 3 Yes Votes
• 2 Yes Votes, 1 No Vote
• 1 Yes Vote, 2 No Vote
• 3 No Votes

28
Preference Ordering

• No gain for individual in voting No as the vote
  is anonymous
• 1 Yes, 2 No Worst for all no raise Yes
  vote undermines legislators claim to have voted
  no
• 3 No No raise, but i, j, ks claim to have
  voted No now credible.

29
Preference Ordering

• 3 Yes Public angry, but raise passes.
  Yippee!
• 2 Yes, 1 No Raise passes, but now i, j, k
  can all claim to have voted No (Best outcome)
  do voters believe?

30
Secret Vote Outcome
31
Secret Vote Outcome
32
Secret Vote Outcome
i has incentive to switch vote.
33
Secret Vote Outcome
Nobody has incentive to switch vote from here.
34
Secret Vote Outcome
Thus, this is the Nash equilibrium
35
Secret Vote Outcome
But so is this
36
Secret Vote Outcome
And so is this
37
Conclusions

• Under secret ballot, game has three equilibria
• Which emerges?
• Will i, j, k fail to coordinate?
• Will they be able to commit to their bargain?

38
Conclusions

• Results rest on arguments about preferences
• Different preferences, different outcome
• What if the legislator who voted No in Nash
  received extra benefit?