<?xml version="1.0" ?><html>
<head>
<title></title>
</head>
<body>
<div align="left"><font face="Courier New"><span style="font-size:12pt">Hi,</span></font></div>
<div align="left"><br/>
</div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">Jobst wrote:</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> Steve wrote:</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">-snip-</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">>> Jobst, does a person behave differently when forced</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">>> to choose between</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
two alternatives about which he is</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">>> undecided than when forced to</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
choose between two he</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">>> believes are equivalent? Why should the</span></font><font face="Courier New" color="#0000ff"><span
style="font-size:12pt"> distinction</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">>> affect the design of the voting method?</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">-snip-</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">></span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> Well, again my favourite example: Sincere preferences </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> A>B, C>D, no</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
further preferences. So here I'm undecided</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> between A and C, A and D, B</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
and C, and B and D. If I</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> would consider A and C equivalent instead, I</span></font><font face="Courier New" color="#0000ff"><span
style="font-size:12pt"> would </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> most probably also have the preferences A>D and C>B </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> since usually</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
my preferences are transitive. The</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> difference shows up easily in the</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
Hasse diagrams </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> of the corresponding preference quasi-orders:</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> </span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> undecidedness: equivalence:</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> A C
A=C</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> | |
/ \</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> | | /
\</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> B D B
D</span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> </span></font></div>
<div align="left"><font face="Courier New" color="#0000ff"><span style="font-size:12pt">> When a voters has no undecidedness and his preferences </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> are transitive,</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
she has a linear ordering in which</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> the possible equivalences are the</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
ties. When she has</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> some undecided pairs, then she does not have a linear</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> ordering but only a partial ordering (when there </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> are no equivalences)</span></font><font face="Courier New" color="#0000ff"><span style="font-size:12pt">
or a quasi-order (when there </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> are also equivalences). That's all basic</span></font><font face="Courier New" color="#0000ff"><span
style="font-size:12pt"> order </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">> theory, isn't it?</span></font></div>
<div align="left"><br/></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">Well, first a minor point... I believe the word </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">"linear" is not used when there are equivalences.</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">A linear ordering means the same as a "strict" </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">ordering. When there are equivalences, the term </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">in the literature is usually <b>"</b>weak" ordering. </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">(I sometimes call a weak ordering a nonstrict </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">ordering, or just an ordering.) </span></font></div>
<div align="left"><br/>
</div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">It may be misleading to say that because voters </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">tend to have transitive preferences that it's </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">reasonable to assume when A?C is changed to A=C </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">that it will also lead to A>D and C>B. So, </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">to avoid confusing the issues when studying </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">the difference between undecidedeness and </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">equivalence on voter behavior, I think we</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">should prefer an example such as this:</span></font></div>
<div align="left"><br/>
</div>
<div align="left"><font face="Courier New"><span style="font-size:12pt"> undecidedness equivalence</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt"> A C
A=C</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt"> |\ /|
|</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt"> | \/ |
|</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt"> | /\ |
|</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt"> |/ \|
|</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt"> B D
B=D</span></font></div>
<div align="left"><br/>
</div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">Would a voter behave differently given the </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">undecidedness preferences on the left instead </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">of the equivalence preferences on the right?</span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">How would the undecided voter behave if allowed </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">to express equivalence but not undecidedness </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">when voting? Is the distinction significant </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">enough to make it worth complicating the voting </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">method? I'm sorry I didn't make my questions </span></font></div>
<div align="left"><font face="Courier New"><span style="font-size:12pt">clear earlier and I hope they are clearer now.</span></font></div>
<div align="left"><br/></div>
<div align="left"><br/>
</div>
<div align="left"><font face="Courier"><span style="font-size:12pt">-- Steve</span></font></div>
</body>
</html>