I'm resubmitting this in a text-friendly format, at Forest's request. I'll also take the opportunity to add one paragraph about how rated methods can fail to find the highest-utility candidates in scenarios like this. Added text is marked ADDED.<br>
<br><div class="gmail_quote">---------- Forwarded message ----------<br>From: <b class="gmail_sendername">Jameson Quinn</b> <span dir="ltr"><<a href="mailto:jameson.quinn@gmail.com">jameson.quinn@gmail.com</a>></span><br>
Date: 2011/12/25<br>Subject: SODA, negotiation, and weak CWs<br>To: EM <<a href="mailto:election-methods@lists.electorama.com">election-methods@lists.electorama.com</a>><br><br><br>In order to have optimum Bayesian Regret, a voting system should be able to not elect a Weak Condorcet Winner (WCW), that is, a CW whose utility is lower than the other candidates. Consider the following payout matrices:</div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Group Size Candidate Utilities</font></div><div class="gmail_quote"><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Scenario 1 (zero sum) A B C</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">1</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total utility</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span></div>
<div class="gmail_quote"><span class="Apple-style-span" style="font-family:'courier new',monospace"><br></span></div><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Scenario 2 (pos. sum) </font><span class="Apple-style-span" style="font-family:'courier new',monospace">A</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">B</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">C</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">1</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">3</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">1.5</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">2</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">3</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total </font><span class="Apple-style-span" style="font-family:'courier new',monospace">utility</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">12</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">12</span></div>
<div class="gmail_quote"><span class="Apple-style-span" style="font-family:'courier new',monospace"><br></span></div><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Scenario 3 (neg. sum) A</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">B</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">C</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">4</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">3</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">2</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">1</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total </font><span class="Apple-style-span" style="font-family:'courier new',monospace">utility</span><font class="Apple-style-span" face="'courier new', monospace"> </font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace"> 16</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">11</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">16</font></div>
</div><div class="gmail_quote"><br></div><div class="gmail_quote"><br></div><div class="gmail_quote">All three scenarios consist of 3 groups of voters: groups a, b, and c, with 4, 2, and 3 voters respectively, for a total of 9 voters. All scenarios have 3 candidates: A, B, and C, who favor their respective groups. And in all three scenarios, candidate B is the CW, because the preference matrix is always<div>
<div style="background-color:transparent"><span>
<div><br></div><div>4: A>B</div><div>2: B>C</div><div>3: C>B</div><div><br></div><div>But in scenario 1, the utilities of the three candidates are balanced; in scenario 2, B has the highest utility; and in scenario 3, A and C have the highest utilities.</div>
<div><br></div><div>Obviously, any purely preferential system will tend to give the same result in all three scenarios. This might not be 100% true if strategy propensity depended on the utility payoff of a strategy; but the strategic possibilities would have to be just right for a method to "get it right" for this reason.</div>
<div><br></div><div>It's easy to see how Range could "get it right" in scenarios 2 and 3. With just a bit of strategy, it's also easy to see how it could successfully find the CW in scenario 1.</div><div>
<br></div><div>You can also construct plausible stories of how Approval or MJ could "get it right" in all 3 scenarios, although it probably involves adding some random noise to voting patterns rather than assuming pure "honest" votes.</div>
<div><br></div><div>ADDED: Of course, Range, Approval, and MJ can all get these scenarios "wrong" too. Because the scenarios present a classic chicken dilemma between B and C, these rated systems could all end up electing A, regardless of utility.</div>
<div><br></div><div>But what about SODA? As a primarily preferential system, it seems that it should give the same result in all three scenarios. If candidates all rationally pursue the interests of their primary constituency, then A will approve B to prevent B from having to approve C, leaving a win for B.</div>
<div><br></div><div>But if candidate A decides to make an ultimatum, things could go differently. A says to B: "Make some promise that transfers 0.5 point of utility to each member of group a, or I will not approve you." Assume that B can make a promise to transfer utility from one group to another at 80% efficiency; and that such promises are not strictly enforceable. Thus, if A gets too greedy, B can simply promise the moon and not keep the promise; but if A asks for something reasonable, B will see honesty as worth it.</div>
<div><br></div><div>B could promise to transfer 0.5 point of utility from groups b and c to group a. Since utility transfers are assumed to be only 80% efficient, that transfer of 2.5 utility points would result in a net loss of 0.5. So the payoffs would be:</div>
<div><div><div style="background-color:transparent"><span><div style="font-weight:bold"><div class="gmail_quote" style="font-weight:normal"><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace"><br class="Apple-interchange-newline">
Group Size Candidate Utilities</font></div><div class="gmail_quote"><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Scenario 1a(zero sum) A B C</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">1.5</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> 2.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> 1.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total utility</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">15.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span></div>
<div><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace"><br class="Apple-interchange-newline">Group Size Candidate Utilities</font></div><div class="gmail_quote">
<div class="gmail_quote">
<font class="Apple-style-span" face="'courier new', monospace">Scenario 1b(zero sum) A B C</font></div><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">1.5 </font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> 3 </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> 1.1</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total utility</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">15.3</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">16</span></div>
<div><span class="Apple-style-span" style="font-family:'courier new',monospace"><br></span></div></div></div></div></div><div class="gmail_quote" style="font-weight:normal"><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Scenario 2a(pos. sum) </font><span class="Apple-style-span" style="font-family:'courier new',monospace">A</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">B</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">C</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">1.5</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> 2.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">1.5</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> 1.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">3</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total </font><span class="Apple-style-span" style="font-family:'courier new',monospace">utility</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">12</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">15.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">12</span></div>
<div class="gmail_quote"><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace"><br class="Apple-interchange-newline">Scenario 3a(neg. sum) A</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">B</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">C</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">4</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> 1 </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> 2.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">2</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> 0.5</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total </font><span class="Apple-style-span" style="font-family:'courier new',monospace">utility</span><font class="Apple-style-span" face="'courier new', monospace"> </font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace"> 16</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">10.5</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">16</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace"><br class="Apple-interchange-newline">Scenario 3b(neg. sum) A</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">B</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">C</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">a</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">4</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> 1 </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">b</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">2</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">0</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> 3 </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace">2</span></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">c</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">3</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">0</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> 0.1</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">4</font></div>
<div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace">Total </font><span class="Apple-style-span" style="font-family:'courier new',monospace">utility</span><font class="Apple-style-span" face="'courier new', monospace"> </font><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace"> 16</font><span class="Apple-style-span" style="font-family:'courier new',monospace"> 10.3</span><span class="Apple-style-span" style="font-family:'courier new',monospace"> </span><font class="Apple-style-span" face="'courier new', monospace">16</font></div>
</div><div class="gmail_quote"><font class="Apple-style-span" face="'courier new', monospace"><br></font></div></div></div><div>Note that in scenarios 1a and 2a, this utility transfer has left B giving the same utility to groups a and c, while in scenario 3a, B has switched from favoring group c over group a, to favoring group a over group c. Also, note that in scenario 2a, group b still gets a full point of advantage with candidate B versus what they would get with candidate C, whereas in the other two Xa scenarios, group b only gets half a point of advantage there. If group b demands a full point of advantage, then B could only meet the ultimatum in scenario 1 by taking all the utility from group c, as in scenarios 1b and 3b. Again, this would leave c with less utility than a.</div>
<div><br></div><div>I believe that these factors tend to make it more likely that B would meet the ultimatum in scenario 2 than in the other scenarios (because they'd be reluctant to anger group c by "unfairly" favoring group a). Of course, A could realize this, and simply not attempt to make the ultimatum in scenarios 1 and 3; and then B would still win. But A's utility payouts show that they honestly have no preference between groups b and c, so I think that it is not unreasonable to imagine that they'd make the ultimatum in all three cases.</div>
<div><br></div><div>The upshot is, there is a plausible (though perhaps not too likely) mechanism for SODA to avoid electing a CW specifically in cases where that CW is intrinsically weak. And that's with perfect information; I'd argue that the mechanism would work even better in cases where B's strength were illusory; that is, where groups a and c were overestimating their payoff from candidate B because of the A/C rivalry. Candidate A, realizing that they were choosing between B and C, would be more careful about assessing the relative payoffs between those candidates than group a, distracted by the A/C rivalry, had been.</div>
<div><br></div><div>Thoughts?</div><div><br></div><div>Jameson</div></span></div></div></div></span></div></div>
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