Chapter Thirty-One

1
Chapter Thirty-One

• Welfare

2
Social Choice

• Different economic states will be preferred by
  different individuals.
• How can individual preferences be aggregated
  into a social preference over all possible
  economic states?

3
Aggregating Preferences

• x, y, z denote different economic states.
• 3 agents Bill, Bertha and Bob.
• Use simple majority voting to decide a state?

4
Aggregating Preferences
More preferred
Less preferred
5
Aggregating Preferences
Majority Vote Results
x beats y
6
Aggregating Preferences
Majority Vote Results
x beats y y beats z
7
Aggregating Preferences
Majority Vote Results
x beats y y beats z z beats x
8
Aggregating Preferences
Majority Vote Results
No socially best alternative!
x beats y y beats z z beats x
9
Aggregating Preferences
Majority Vote Results
No socially best alternative!
x beats y y beats z z beats x
Majority voting does not always
aggregate transitive individual preferences into
a transitive social preference.
10
Aggregating Preferences
11
Aggregating Preferences
Rank-order vote results (low score wins).
12
Aggregating Preferences
Rank-order vote results (low score wins).
x-score 6
13
Aggregating Preferences
Rank-order vote results (low score wins).
x-score 6 y-score 6
14
Aggregating Preferences
Rank-order vote results (low score wins).
x-score 6 y-score 6 z-score 6
15
Aggregating Preferences
Rank-order vote results (low score wins).
No state is selected!
x-score 6 y-score 6 z-score 6
16
Aggregating Preferences
Rank-order vote results (low score wins).
No state is selected!
x-score 6 y-score 6 z-score 6
Rank-order voting is indecisive in this case.
17
Manipulating Preferences

• As well, most voting schemes are manipulable.
• I.e. one individual can cast an untruthful vote
  to improve the social outcome for himself.
• Again consider rank-order voting.

18
Manipulating Preferences
These are truthful preferences.
19
Manipulating Preferences
These are truthful preferences. Bob introduces
a new alternative
20
Manipulating Preferences
These are truthful preferences. Bob introduces
a new alternative
21
Manipulating Preferences
These are truthful preferences. Bob introduces
a new alternative and then lies.
22
Manipulating Preferences
These are truthful preferences. Bob introduces
a new alternative and then lies.
Rank-order vote results.
x-score 8
23
Manipulating Preferences
These are truthful preferences. Bob introduces
a new alternative and then lies.
Rank-order vote results.
x-score 8 y-score 7
24
Manipulating Preferences
These are truthful preferences. Bob introduces
a new alternative and then lies.
Rank-order vote results.
x-score 8 y-score 7 z-score 6
25
Manipulating Preferences
These are truthful preferences. Bob introduces
a new alternative and then lies.
Rank-order vote results.
x-score 8 y-score 7 z-score 6 ?-score 9
z wins!!
26
Desirable Voting Rule Properties

• 1. If all individuals preferences are complete,
  reflexive and transitive, then so should be the
  social preference created by the voting rule.
• 2. If all individuals rank x before y then so
  should the voting rule.
• 3. Social preference between x and y should
  depend on individuals preferences between x and
  y only.

27
Desirable Voting Rule Properties

• Kenneth Arrows Impossibility Theorem The only
  voting rule with all of properties 1, 2 and 3 is
  dictatorial.

28
Desirable Voting Rule Properties

• Kenneth Arrows Impossibility Theorem The only
  voting rule with all of properties 1, 2 and 3 is
  dictatorial.
• Implication is that a nondictatorial voting rule
  requires giving up at least one of properties 1,
  2 or 3.

29
Social Welfare Functions

• 1. If all individuals preferences are complete,
  reflexive and transitive, then so should be the
  social preference created by the voting rule.
• 2. If all individuals rank x before y then so
  should the voting rule.
• 3. Social preference between x and y should
  depend on individuals preferences between x and
  y only.

30
Social Welfare Functions

• 1. If all individuals preferences are complete,
  reflexive and transitive, then so should be the
  social preference created by the voting rule.
• 2. If all individuals rank x before y then so
  should the voting rule.
• 3. Social preference between x and y should
  depend on individuals preferences between x and
  y only.

Give up which one of these?
31
Social Welfare Functions

• 1. If all individuals preferences are complete,
  reflexive and transitive, then so should be the
  social preference created by the voting rule.
• 2. If all individuals rank x before y then so
  should the voting rule.
• 3. Social preference between x and y should
  depend on individuals preferences between x and
  y only.

Give up which one of these?
32
Social Welfare Functions

• 1. If all individuals preferences are complete,
  reflexive and transitive, then so should be the
  social preference created by the voting rule.
• 2. If all individuals rank x before y then so
  should the voting rule.

There is a variety of voting procedures with both
properties 1 and 2.
33
Social Welfare Functions

• ui(x) is individual is utility from overall
  allocation x.

34
Social Welfare Functions

• ui(x) is individual is utility from overall
  allocation x.
• Utilitarian

35
Social Welfare Functions

• ui(x) is individual is utility from overall
  allocation x.
• Utilitarian
• Weighted-sum

36
Social Welfare Functions

• ui(x) is individual is utility from overall
  allocation x.
• Utilitarian
• Weighted-sum
• Minimax

37
Social Welfare Functions

• Suppose social welfare depends only on
  individuals own allocations, instead of overall
  allocations.
• I.e. individual utility is ui(xi), rather than
  ui(x).
• Then social welfare iswhere is an
  increasing function.

38
Social Optima Efficiency

• Any social optimal allocation must be Pareto
  optimal.
• Why?

39
Social Optima Efficiency

• Any social optimal allocation must be Pareto
  optimal.
• Why?
• If not, then somebodys utility can be increased
  without reducing anyone elses utility i.e.
  social suboptimality ? inefficiency.

40
Utility Possibilities
OB
0
0
OA
41
Utility Possibilities
OB
0
0
OA
42
Utility Possibilities
OB
0
0
OA
43
Utility Possibilities
OB
0
0
OA
44
Utility Possibilities
OB
0
0
OA
45
Utility Possibilities
OB
0
0
OA
46
Utility Possibilities
Utility possibility frontier (upf)
OB
0
0
OA
47
Utility Possibilities
Utility possibility frontier (upf)
OB
0
0
Utility possibility set
OA
48
Social Optima Efficiency
Upf is the set of efficient
utility pairs.
49
Social Optima Efficiency
Upf is the set of efficient
utility pairs.
Social indifference curves
50
Social Optima Efficiency
Upf is the set of efficient
utility pairs.
Higher social welfare
Social indifference curves
51
Social Optima Efficiency
Upf is the set of efficient
utility pairs.
Higher social welfare
Social indifference curves
52
Social Optima Efficiency
Upf is the set of efficient
utility pairs.
Social optimum
Social indifference curves
53
Social Optima Efficiency
Upf is the set of efficient
utility pairs.
Social optimum is efficient.
Social indifference curves
54
Fair Allocations

• Some Pareto efficient allocations are unfair.
• E.g. one consumer eats everything is efficient,
  but unfair.
• Can competitive markets guarantee that a fair
  allocation can be achieved?

55
Fair Allocations

• If agent A prefers agent Bs allocation to his
  own, then agent A envies agent B.
• An allocation is fair if it is
• Pareto efficient
• envy free (equitable).

56
Fair Allocations

• Must equal endowments create fair allocations?

57
Fair Allocations

• Must equal endowments create fair allocations?
• No. Why not?

58
Fair Allocations

• 3 agents, same endowments.
• Agents A and B have the same preferences. Agent
  C does not.
• Agents B and C trade ? agent B achieves a more
  preferred bundle.
• Therefore agent A must envy agent B ? unfair
  allocation.

59
Fair Allocations

• 2 agents, same endowments.
• Now trade is conducted in competitive markets.
• Must the post-trade allocation be fair?

60
Fair Allocations

• 2 agents, same endowments.
• Now trade is conducted in competitive markets.
• Must the post-trade allocation be fair?
• Yes. Why?

61
Fair Allocations

• Endowment of each is
• Post-trade bundles are
  and

62
Fair Allocations

• Endowment of each is
• Post-trade bundles are
  and
• Thenand

63
Fair Allocations

• Suppose agent A envies agent B.
• I.e.

64
Fair Allocations

• Suppose agent A envies agent B.
• I.e.
• Then, for agent A,

65
Fair Allocations

• Suppose agent A envies agent B.
• I.e.
• Then, for agent A,
• Contradiction. is not affordable
  for agent A.

66
Fair Allocations

• This proves If every agents endowment is
  identical, then trading in competitive markets
  results in a fair allocation.

67
Fair Allocations
OB
OA
Equal endowments.
68
Fair Allocations
OB
Given prices p1 and p2.
Slope -p1/p2
OA
69
Fair Allocations
OB
Given prices p1 and p2.
Slope -p1/p2
OA
70
Fair Allocations
OB
Given prices p1 and p2.
Slope -p1/p2
OA
71
Fair Allocations
OB
Post-trade allocation -- is it fair?
OA
72
Fair Allocations
OB
Swap As and Bs post-trade allocations
Post-trade allocation -- is it fair?
OA
73
Fair Allocations
OB
Swap As and Bs post-trade allocations
Post-trade allocation -- is it fair?
OA
A does not envy Bs post-trade allocation. B does
not envy As post-trade allocation.
74
Fair Allocations
OB
Swap As and Bs post-trade allocations
Post-trade allocation -- is it fair?
OA
Post-trade allocation is Pareto-efficient
and envy-free hence it is fair.