[EM] Some interesting example

[Jobst Heitzig]

Hi folks!

Pretty long gap since the last posting. Perhaps some interesting example
would make a nice restart... Here's some example I came upon yesterday
when thinking about "Independence from Pareto-dominated alternatives"
and "Independence from Pareto-dominated alternatives", two criteria
which distinguish between Ranked Pairs, Beatpath, and River. I think it
would be interesting to discuss whether A or B should win in this case:


Sincere Preferences:
--------------------
	16 B>D>A>C
	17 D>A>B>C
	34 A>B>C>D
	33 B>C>D>A


Defeats: A>B>C>D, A>C, B>D  ==> Smith Set = {A,B,C,D}
--------


Copeland Scores:  A  B  C  D
----------------  1  1  2  2


Preference Matrix:
------------------
	...>... A    B    C    D
	A            51   67  (34)
	B      (49)      100   83
	C      (33)  (0)       67
	D       66  (17) (33)


Beatpath Matrix:
----------------
	...>... A    B    C    D
	A            51   67   67
	B       66       100   83   ==> B wins in Beatpath
	C       66   51        67
	D       66   51   66

==> Immune Set = {A,B}


First Choice Support:   A  B  C  D
---------------------  34 49  0 17


Sincere Winners:
----------------
Plurality		B
IRV			A  (17 transferred from D to A)
Borda			B  (scores 0..3 give A 152, B 232, C 100, D 116)
Minimax,SD,Beatpath	B  (drop only A>B)
Kemeny			A  (order A>B>C>D fits best)
Ranked Pairs		A  (affirms B>C,B>D,A>C>D,A>B, giving A>B>C>D)
River			B  (affirms B>C,B>D>A)
Approval		B  (see below)
Grand compromise	B  (see below)

Sincere Second Winners:
-----------------------
These win in the sincere situation when the above winner is removed:

Plurality		A!
IRV			B
Borda			A! (scores 0..2 give A 101, C 100, D 99)
Minimax,SD,Beatpath	A! (drop D>A)
Kemeny			B  (order B>C>D fits best)
Ranked Pairs		B  (affirms B>C,B>D,C>D, giving B>C>D)
River			A! (affirms A>C>D)
Approval		A!
Grand compromise	A!

Hence Plurality, Borda, Minimax, SD, Beatpath, River, Approval, and the
recent grand compromise proposal all fail
"Immunity from second place complaints".


Winners when Pareto-dominated C is removed:
-------------------------------------------
Plurality		B
IRV			A  (17 transferred from D to A)
Borda			B  (scores 0..2 give A 85, B 132, D 83)
Minimax,SD,Beatpath	B  (drop A>B)
Kemeny			B! (order B>D>A fits best)
Ranked Pairs		B! (affirms B>D>A)
River			B  (affirms B>D>A)
Approval		B
Grand compromise	B

Hence Kemeny and Ranked Pairs fail
"Independence from Pareto-dominated alternatives".


Approval:
---------
Starting with
	16 B	(>D>A>C) ("group BD")
	17 D	(>A>B>C) ("group D")
	34 A	(>B>C>D) ("group A")
	33 B	(>C>D>A) ("group BC")
would give
	A 34, B 49, C 0,  D 17, hence B would win.
Then group D has incentive to switch to
	17 D,A
giving
	A 51, B 49, C 0,  D 17, hence A would win.
But then groups BD and BC have incentive to vote
	16 B,D
	33 B,C,D
giving
	A 51, B 49, C 33, D 66, hence D would win.
Now group A has incentive to switch to
	34 A,B
giving
	A 51, B 83, C 33, D 66, hence B would win.
Now in this situation
	16 B,D  (>A>C) ("group BD")
	17 D,A  (>B>C) ("group D")
	34 A,B  (>C>D) ("group A")
	33 B,C,D  (>A) ("group BC")
still all vote sincerely and no-one has an incentive to vote anything
else, whether sincerely or insincerely.

Hence B is both a sincere and an equilibrium winner in Approval.

All other methods mentioned above have no equilibria since all of them
fulfil the majority criterion and thus have no equilibria when no
sincere CW exists.


Grand compromise method:
------------------------
Recall that this is a combination of River and Approval.
In the example, we find the same "sincere equilibrium" as in Approval:

When we denote the approval cutoff by >> and start with
	16 B >> D>A>C ("group BD")
	17 D >> A>B>C ("group D")
	34 A >> B>C>D ("group A")
	33 B >> C>D>A ("group BC")
we get the following defeat strengths:
	...>... A    B    C    D
	A           34   34    -
	B       -        49   49
	C       -    -         0
	D      17    -    -
Using the River technique to resolve cycles, we would affirm B>C,B>D,A>B
so that A would win.
But then groups BD and BC have incentive to switch to
	16 B>D >> A>C
	33 B>C>D >> A
giving defeat strengths
	...>... A    B    C    D
	A           34   34    -
	B       -        16    0
	C       -    -         0
	D      66    -    -
so that D>A>B,A>C would be affirmed and D would win.
Now group A has incentive to switch to
	34 A>B >> C>D
giving defeat strengths
	...>... A    B    C    D
	A            0   34    -
	B       -        50   34
	C       -    -         0
	D      66    -    -
so that D>A,B>C,B>D would be affirmed and B would win.
Now in this situation
	16 B>D >> A>C	("group BD")
	17 D >> A>B>C 	("group D")
	34 A>B >> C>D	("group A")
	33 B>C>D >> A	("group BC")
still all vote only sincere preferences and no-one has an incentive to
specify any other approval cutoffs.