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\newcommand{\fpause}{\pause\vfill}
\newcommand{\crit}[1]{{\bf #1}}

\title[]{
Voting: How it works, and why it doesn't}
\author[Staecker]{Chris Staecker}
\institute[Fairfield]{Fairfield University}
\date[]{Fairfield U. Mathematics \& Computer Science Colloquium \\ Election Day 2012}

\begin{document}
\frame{\titlepage}

\section{Introduction}
\frame{
This talk is about fairness in voting systems.
\fpause
I'll discuss specifically the unfairness in our system of voting.
}

\frame{
\frametitle{I don't mean...}
\pause
I will not talk about actual cheating
\fpause
These are real issues, but I want to talk about fairness \emph{of the counting system} \pause assuming that everybody is following the rules.
\fpause
I also will not discuss the electoral college.
\fpause
This is a crazy overlay onto our basic voting system which makes everything slightly weirder.
}

\frame{
I'm interested in the system at a much more fundamental level. \fpause Just the basic idea of counting up votes and deciding the winner.

\fpause
This turns out to be much more complicated than you might expect.
\fpause
Actually, voting is an insane idea when you think about it.
}

\frame{
Imagine a bunch of people disagree about something. \pause How will we decide?
\fpause
Let's just ask everybody what their opinion is, \pause then combine all these answers into a single ``will of the people''.
\fpause
This sounds sketchy.
}

\frame{
Something that complicates everything:
\begin{center}
\emph{Preferences of groups of people do not behave like preferences of individual people.}
\end{center}
\fpause
This is the \emph{Condorcet paradox}.

(Condorcet, 1743-1794)
}

\frame{
\frametitle{Condorcet paradox}
\begin{center}
\emph{Preferences of groups of people do not behave like preferences of individual people.}
\end{center}
\fpause
Imagine an election with three candidates $A$, $B$, $C$.
\fpause
No person would ever say: ``I like $A$ more than $B$, and $B$ more than $C$, and $C$ more than $A$''.
\fpause
Individual preferences are \emph{transitive}.
}

\frame{
But let's ask a group of people to rank their choices, and imagine they say:
\begin{center}\begin{tabular}{ccc}
15 & 11 & 13 \\
\hline 
A & B & C \\
B & C & A \\
C & A & B 
\end{tabular}\end{center}

\fpause
Here, 72\% prefer $A$ over $B$. \pause

67\% prefer $B$ over $C$. \pause

62\% prefer $C$ over $A$.
\fpause
So what is the ``will of the people''?
\fpause
Sounds like there is no coherent will of the people.
}

\frame{
Major goal for the talk:
\fpause
Various different ways to look at preferences and decide the winner. \pause Which is the best?
\fpause
Basically, a winner-selection method should analyse the preferences, and choose a winner based on some relevant details of the set of preferences.
\fpause
For a reasonably fair system:
\begin{itemize}
\item If the society actually has a uniform preference, the decision should reflect this.\pause
\item The decision should not depend on irrelevant details of the preferences.
\end{itemize}
}

\section{Let's vote!}

\frame{
\frametitle{Let's vote!}
\fpause
Voting for US president is boring. \pause We will vote for:
\fpause
The second-best bounty hunter from \emph{The Empire Strikes Back}
\[ \includegraphics[scale=.3]{bountyhunters.jpg} \]
\fpause
Obviously Boba Fett is the best. \pause We'll vote for second best.
}

\newcommand{\bhpic}[2]{%
\centerline{\includegraphics[height=50px, width=50px]{#1.jpg}}
\centerline{\only<#2->{#1}}\vspace*{\fill}
}

\frame{
Here are the choices:
\vfill
\begin{columns}[t]
\begin{column}{5cm}
\bhpic{Bossk}2

\bhpic{Zuckuss}3

\end{column}
\begin{column}{5cm}
\bhpic{4-LOM}4

\bhpic{Dengar}5

\end{column}
\end{columns}
\bhpic{IG-88}6
\pause\pause\pause\pause\pause
}

\frame{
To make it interesting, let's \emph{rank} our choices.
\fpause
Choose your \#1, \#2, etc. choice. 
\fpause
After we vote, we'll count up the votes and have our decision.
}

\frame{
Your ballot will look like this:
\[ \includegraphics[scale=.3]{ballot.jpg} \]
}

\frame{
Vote on the tablets going around, or:
\vfill
Connect to the ``staecker'' wi-fi network, and visit:

{\tt http://staecker.local/vote}
}

\frame{
Once we have all the votes, we'll tally them in the obvious way.
\fpause
Actually there is no obvious way.
\fpause
There are lots and lots of \emph{winner selection methods} that we could use.
\fpause
Even reasonable alternative systems will produce wildly different outcomes.
}

\section{8 voting methods}
\frame{
Stalin (1920s): ``I consider it completely unimportant who in the party will vote, or how; but what is extraordinarily important is this Ñ who will count the votes, and how.''
\fpause
Here comes 8 different winner selection methods for ranked ballots.
}

\subsection{Plurality}
\frame{
\frametitle{Plurality}
This is basically what we use in USA. 
\fpause
Whoever gets the most first place votes is the winner.
\fpause
All rankings except first place are ignored.
}

\subsection{Anti-plurality}
\frame{
\frametitle{Anti-plurality}
A silly variation on plurality:
\pause 

whoever gets the fewest last-place votes is the winner.
\fpause
This will elect the least-bad candidate, rather than the most-good.
\fpause
Use this in a ``lesser of evils'' election.
}

\begin{comment}
\subsection{Bucklin}
\frame{
\frametitle{Bucklin voting}
Count only the total of the first place votes. \pause 

If somebody has a majority, they are the winner.
\fpause
If not, add in the second place votes to the earlier totals. \pause

If a majority is reached, the highest total is the winner.
\fpause
If not, continue with third place votes, etc.
}

\frame{
So for this one:
\[ \begin{tabular}{cccc}
1 & 3 & 2 & 4 \\
\hline
A & B & C & A \\
B & A & A & C \\
C & C & B & B 
\end{tabular} \]
\pause

First place totals are:
\[ A:5 \quad B:3 \quad C:2 \]
nobody has a majority ($>$5). 
\fpause

Add in the second place votes too:
\[ A: 10 \quad B: 7 \quad C: 6 \]
so $A$ wins.
}
\end{comment}

\subsection{Borda count}
\frame{
\frametitle{Borda count}
Everybody gets points:
\fpause
for $n$ candidates:\pause
\begin{itemize}
\item a first place vote is worth $n$ points \pause
\item a second place vote is worth $n-1$ points \pause
\item \dots
\item a last place vote is worth 1 point
\end{itemize}
}

\frame{
\frametitle{Borda count}
So if the candidates are $A,B,C$ and the votes are like this:
\[ \begin{tabular}{cccc}
1 & 3 & 2 & 4 \\
\hline
A & B & C & A \\
B & C & A & C \\
C & A & B & B 
\end{tabular} \]
\fpause
$A$ gets: $1 \times 3 + 3 \times 1 + 2 \times 2 + 4 \times 3 = 22$ points 

\pause
$B$ gets: $1 \times 2 + 3 \times 3 + 2 \times 1 + 4 \times 1 = 17$ points

\pause
$C$ gets: $1\times 3 + 3 \times 2 + 2\times 3 + 4\times 2 = 27$ points
\fpause
$C$ wins.
}

\begin{comment}
\frame{
(by the way)

\begin{align*}
1 \times 3 + 3 \times 1 + 2 \times 2 + 4 \times 3 &= 22 \\
1 \times 2 + 3 \times 3 + 2 \times 1 + 4 \times 1 &= 17 \\
1\times 1 + 3 \times 2 + 2\times 3 + 4\times 2 &= 27
\end{align*}
\fpause

is nicer like this:\fpause
\[ \begin{bmatrix}
3 & 1 & 2 & 3 \\
2 & 3 & 1 & 1 \\
1 & 2 & 3 & 2 \end{bmatrix}
\begin{bmatrix} 1 \\ 3 \\ 2 \\ 4 \end{bmatrix}
= \begin{bmatrix} 22 \\ 17 \\ 27 \end{bmatrix}
\]
}
\end{comment}

\subsection{Instant runoff}
\frame{
\frametitle{Instant runoff}
Do several rounds. \pause Each time, eliminate the one with the fewest first-place votes.
\fpause
\[ \begin{tabular}{cccc}
1 & 3 & 2 & 4 \\
\hline
A & B & C & A \\
B & C & A & C \\
C & A & B & B 
\end{tabular} \]
\fpause
In the first round, we eliminate $C$. 
}

\frame{
Eliminating $C$ looks like:
\[ 
\begin{tabular}{cccc}
1 & 3 & 2 & 4 \\
\hline
A & B & C & A \\
B & C & A & C \\
C & A & B & B 
\end{tabular} 
\to
\begin{tabular}{cccc}
1 & 3 & 2 & 4 \\
\hline
A & B & A & A \\
B & A & B & B 
\end{tabular}  \pause
= 
\begin{tabular}{cc}
7 & 3 \\
\hline
A & B \\
B & A
\end{tabular}
\]

\fpause
Now we eliminate $B$ and $A$ wins.
\fpause
This method is used in Australia, Ireland, and a few local elections in US.
}

\subsection{Coombs, Borda-runoff}
\frame{
\frametitle{IRV Variations}
\pause
Coombs: Same as instant runoff, but in each step eliminate the one with the most losing votes.
(instead of the one with the fewest winning votes)
\fpause
Baldwin: In each round, eliminate the one with the lowest Borda score.
\fpause
Are these all equivalent? \pause no
}

\subsection{Pairwise comparison}
\frame{
\frametitle{Pairwise comparisons}
Pit the candidates against each other one-on-one in all possible matchups
\fpause
Whoever wins the most of these wins the election.
}

\subsection{Random dictator}
\frame{
\frametitle{Random dictator}
The craziest of all of these.
\fpause
Choose a single ballot at random, their first-place choice wins the election.
\fpause
Sounds ridiculous because it's nondeterministic
\fpause
But a person with $x\%$ support will win the election with probability $x\%$, which doesn't sound too bad.
}

\frame{
\frametitle{A little digression}
Votes by lottery were common in ancient democracies. \pause In ancient Athens, almost all government offices were filled by lottery.
\fpause
In their view, election by voting favored candidates who were rich, eloquent, and well-known.
\fpause
Aristotle, \emph{Politics}: ``It is accepted as democratic when public offices are allocated by lot; and as oligarchic when they are filled by election.''
\fpause
Voting was not viewed as an important component of democracy. \fpause A true government ``of the people'' should be made up of ordinary people, chosen at random.
}

\section{Results!}
\frame{
\frametitle{Results!}
Let's see the results of our election.
\fpause
Moral of the story: \fpause 
\begin{center}
Different reasonable voting methods produce different outcomes.
\end{center}
}

\section{Fairness}
\frame{
\frametitle{Fairness}
So which method should be used?
\fpause
We need some criteria for judging fairness of the methods.
\fpause
Hopefully we can come up with some basic principles for fairness,\pause\ and choose a system which satisfies them all.
}

\frame{
I've got 3 basic categories for fairness:
\begin{itemize}
\item Preferences: \pause The winner should be ``preferred'' over the losers \fpause
\item Decisions: \pause If someone switches their vote, the election outcome should change ``appropriately''\fpause
\item Honesty: \pause Voters should have no incentive to vote ``dishonestly'' in order to game the system
\end{itemize}
\fpause
Let's talk some specific ways to measure these kinds of fairness.
}

\subsection{Preferences-based fairness}
\frame{
\frametitle{Preferences-based fairness}
For preferences-based fairness, we'll discuss two specific criteria.
\fpause
These are an attempt to define specifically the idea that the winner should be preferred over the losers 
}

\frame{
\frametitle{The majority criterion}
\fpause
\crit{If a majority of people rank candidate $X$ first, then $X$ should win the election.}
\fpause
This is a very reasonable fairness criterion, and is satisfied by the plurality system.
\fpause
Not satisfied by Borda count:
\[
\begin{tabular}{cc}
4 & 3 \\
\hline
A & B \\
B & C \\
C & A
\end{tabular}
\]
\fpause
In the Borda count, $A$ gets 15 and $B$ gets 19.
\fpause
Here, $A$ is ranked first by a majority, but $B$ wins in the Borda count. 
}

\frame{
\frametitle{The Condorcet criterion}
\fpause
\crit{If some candidate wins in every pairwise comparison, then they should win the election.}
\fpause
A candidate like this would be preferred by a majority \emph{when compared individually to anybody else}.
\fpause
Such a candidate is called a \emph{Condorcet winner}.
\fpause
This is also a very reasonable fairness criterion.
}

\frame{
\frametitle{Twiddle-Dee \& Twiddle-Dum}
Let's use the 2000 (G. W. Bush vs Gore) election as an example. 
\fpause 
The votes were very close in Florida, and basically tied otherwise, so the election would be decided by Florida.
\fpause
Here's the final vote totals in Florida:
\fpause
\begin{center}
\begin{tabular}{r|r}
\hline
Bush & 2,912,790 \\ \pause
Gore & 2,912,253 \\ \pause
Nader & 97,488 \\ \pause
Others & 40,579 
\end{tabular}\end{center}
}

\frame{
\begin{center}
\begin{tabular}{r|r}
\hline
Bush & 2,912,790 \\ 
Gore & 2,912,253 \\ 
Nader & 97,488
\end{tabular}\end{center}
Nader is typically described as ``far left'' on most issues, \pause and it's fair to say most of his voters would have preferred Gore over Bush.
\fpause
So if there had been preferences recorded at the ballot, they might've looked like this:
\begin{center}
\begin{tabular}{ccc}
2,912,790 & 2,912,253 & 97,488 \\
\hline
B & G & N \\
G & B & G \\
N & N & B
\end{tabular}\end{center}
}

\frame{
\begin{center}
\begin{tabular}{ccc}
2,912,790 & 2,912,253 & 97,488 \\
\hline
B & G & N \\
G & B & G \\
N & N & B
\end{tabular}\end{center}
\pause
Here, Gore is a Condorcet winner.
\fpause
But Bush is the plurality winner.
\fpause
The plurality system does not satisfy the Condorcet criterion. \fpause
\[ \includegraphics[scale=.5]{condorcet.PNG} \]
}

\subsection{Decisions-based fairness}
\frame{
\frametitle{Decisions-based fairness}
Let's discuss two criteria related to decision-making.
\fpause
We'll formalize the idea that if someone switches their vote, the election outcome should change ``appropriately''
}

\frame{
\frametitle{Monotonicity}
``Monotonicity'' is a mathematical word meaning that ``things move in the same direction''.
\fpause
\crit{If somebody changes their vote to boost $X$'s ranking without changing the others' relative rankings, this should not hurt $X$.}
\fpause
(this should never cause $X$ to switch from winning to losing)
\fpause
This is satisfied by plurality and Borda count, so they seem pretty fair.
}

\frame{
\frametitle{Irrelevant Alternatives}
\crit{If somebody changes their vote without changing the winner's relative ranking with respect to anybody else, this should not affect the outcome of the election.}
\fpause
Imagine this election between Romney \& Obama, with some third parties: 
\fpause
Say I rank them: Obama, Romney, Johnson, Stein.
\fpause
Say Romney wins, then I say ``wait! I meant Obama, Romney, Stein, Johnson!''
\fpause
This is an ``irrelvant alternative''.
\fpause
In a fair system, this kind of change should not affect the election results.
}

\frame{
Again this sounds like a reasonable criterion for fairness.
\fpause
But the plurality system does not satisfy this.
\fpause
\begin{center}
\begin{tabular}{ccc}
2,912,790 & 2,912,253 & 97,488 \\
\hline
B & G & N \\
G & B & G \\
N & N & B
\end{tabular}\end{center}
Bush is the plurality winner.
\fpause
Now if the $NGB$ voters change to $GNB$, this is an irrelevant alternative.
\fpause
But this will cause Gore to become the winner.
\fpause
So the plurality system does not satisfy the irrelevant alternatives criterion.
}

\subsection{Honest-based fairness}
\frame{
\frametitle{Honesty-based fairness}
One more fairness criterion, of the ``Honesty'' type.
\fpause
\crit{A voter should not have any incentive to vote dishonestly}
\fpause
Such a system is called ``strategy-proof''.
\fpause
If your system is not strategy-proof, the voters need to think carefully about voting ``tactically'', rather than voting their true preferences.
}

\frame{
The plurality system fails miserably here.
\fpause
\begin{center}
\begin{tabular}{ccc}
2,912,790 & 2,912,253 & 97,488 \\
\hline
B & G & N \\
G & B & G \\
N & N & B
\end{tabular}\end{center}
\fpause
The Nader voters would have a better outcome if they'd voted for Gore.
\fpause
Their honesty caused Bush to win, which was their last choice.
}

\frame{
This aspect of the plurality system has deep consequences for our whole political structure.
\fpause
Strategy in our system is based fundamentally on avoiding ``vote-splitting''. 
\fpause
A vote for anybody other than the winner is a wasted vote.
\fpause
This makes politicians always claim that they're winning.
\fpause
This makes the two parties indestructible.
}

\frame{
There is a basic principle in political science known as Duverger's Law (1950s): 
\fpause
Any political structure based on plurality will, after sufficient elapsed time, develop into a two-party system.
\fpause
This is true in our world with very few exceptions. (Canada, UK)
}

\begin{comment}
\frame{
\frametitle{Strategy in Borda}
Strategy is a major problem in the Borda count too. 
\fpause
\begin{center}
\begin{tabular}{ccc}
2,912,790 & 2,912,253 & 97,488 \\
\hline
B & G & N \\
G & B & G \\
N & N & B
\end{tabular}\end{center}
\vfill
Knowing $N$ has very little support, the $BGN$ voters will vote $BNG$ to dump Gore's points.
\fpause
For the same reason $GBN$ voters will vote $GNB$.
}

\frame{
Then we'd have:
\begin{center}
\begin{tabular}{ccc}
2,912,790 & 2,912,253 & 97,488 \\
\hline
B & G & N \\
N & N & G \\
G & B & B
\end{tabular}\end{center}
\fpause
The Borda scores are: 
\begin{align*}
B: 11,748,111\\ G: 11,844,525 \\ N: 11,942,550 
\end{align*}
\fpause
Nader wins! (which almost nobody wanted)
\fpause
Borda (1770s): ``My scheme is intended for only honest men.''
}
\end{comment}

\begin{comment}
\frame{
This is closely related to a very bad effect called ``the DH3 pathology''
\fpause
DH3: when voter strategy causes a winner who is least-preferred by every voter.
\fpause
This can happen in the Borda count, and lots of other systems.
}
\end{comment}

\subsection{Summary}
\frame{
\frametitle{Criteria summary}

This can all be worked out:
\newcommand{\yes}{\checkmark}
\newcommand{\no}{$\times$}

\begin{center}
\begin{tabular}{r|ccccc}
& Maj. & Cond. & Mono. & IA & Strategy-proof \\
\hline
Plurality/ Anti-plurality & \yes & \no & \yes & \no & \no \\
%Anti-plurality & \yes & \no & \yes & \no & \no \\
%Bucklin & \yes & \no & \yes & \no & \no \\
Borda & \no & \no & \yes & \no & \no \\
Instant runoff / Coombs & \yes & \no & \no & \no & \no \\
Baldwin & \yes & \yes & \no & \no & \no \\
Pairwise Comparison & \yes & \yes & \yes & \no & \no \\
Random dictator & \no & \no & \yes & \yes & \yes
\end{tabular}
\end{center}

\fpause
By this table, Borda \& Instant runoff look pretty bad 
\fpause
Pairwise Comparison and Random dictator look pretty good!
\fpause
Of course there are other criteria, so this is not the definitive table.
\fpause
And one can discuss the degree of failure on various criteria.
}

\section{Impossibility results}
\frame{
\frametitle{The bad news}
\pause
Is there a voting system that satisfies all of these criteria?
\fpause
No.
\fpause
There are two classic ``impossibility theorems'' which show that no system can obey all of these.
}

\subsection{Arrow's theorem}
\frame{
\frametitle{Arrow's theorem}
Arrow (1950s): No voting system can satisfy the Condorcet criterion and the irrelevant alternatives criterion.
\fpause
(Actually Arrow's original theorem is stronger, but we'll just talk about this version)
\fpause
Bad news for voting in general.
\fpause
When choosing a voting system, we have to decide whether we want Condorcet or IA.\pause\ You can't have both. \pause (Plurality has neither.)
}

\frame{
Remember 30 minutes ago:
\fpause
We want a voting system such that:
\begin{itemize}
\item If the people actually have a uniform preference, the decision should reflect this.
\item The decision should not depend on irrelevant details of the preferences.
\end{itemize}
\fpause
This is impossible.
}

\frame{
Actually it's easy to see why Arrow's theorem is true for a ranked voting system with no ties:
\fpause
Let's assume that there is a system with the Condorcet winner criterion \emph{and} the irrelvant alternatives criterion, and this will lead to a contradiction.
\fpause
Imagine the election:
\begin{center}
\begin{tabular}{ccc}
1 & 1 & 1 \\
\hline
A & B & C \\
B & C & A \\
C & A & B
\end{tabular}
\end{center}
\fpause
All the votes are symmetric- let's imagine that $A$ is chosen as the winner.
}

\frame{
\begin{center}
\begin{tabular}{ccc}
1 & 1 & 1 \\
\hline
A & B & C \\
B & C & A \\
C & A & B
\end{tabular}
\end{center}
$A$ wins.
\fpause
Now if $BCA$ changes to $CBA$, this is an irrelvant alternative. 
\fpause
Since our system obeys the irrelevant alternatives criterion, $A$ will still win in:
\begin{center}
\begin{tabular}{ccc}
1 & 1 & 1 \\
\hline
A & C & C \\
B & B & A \\
C & A & B
\end{tabular}
\end{center}
\fpause
But now $C$ is a Condorcet winner, so $C$ must win because our system obeys the Condorcet criterion.
\fpause
But we just said $A$ wins, so this is a contradiction.
}

\subsection{Gibbard-Satterthwaite theorem}
\frame{
\frametitle{The Gibbard-Satterthwaite theorem}
Another bit of bad news.
\fpause
Gibbard \& Satterthwaite (1970s): For any voting system, one of the following must be true:\pause
\begin{itemize}
\item The system is dictatorial \pause
\item The system is rigged against one of the candidates \pause
\item The system is not strategy-proof
\end{itemize}
\fpause
The first two are obviously unreasonable for real voting systems, so the summary is:
\fpause
No reasonable voting system is strategy-proof.
}

\frame{
Bad news summary:
\fpause
No voting system can be fair with respect to Condorcet winners while correctly disregarding irrelevant alternatives. (Arrow)
\fpause
Voters under any reasonable voting system have an incentive to try to game the system. (Gibbard-Satterthwaite)
\fpause
Note: It's not just that we \emph{haven't yet figured out} how to get around the issues. 
\fpause
They are mathematically unavoidable.
}

\section{What should we do?}
\frame{
\frametitle{So what should we do?}
The concept of perfectly fair voting is logically impossible. \pause So what should we do?
\fpause
No clear answers.
\fpause
Winston Churchill (1947): ``democracy is the worst form of government except all those other forms that have been tried''
\fpause
We shouldn't abandon voting.
}

\frame{
Should we continue to use the plurality system?
\vfill
\only<1->{{\bf Pros}}: \pause Simplicity.\pause\ Easy for voters to understand\pause, easy to tabulate results.
\fpause
\only<1->{{\bf Cons}}: \pause Not Condorcet-fair (etc.)\pause, encourages ``only vote for the winner''\pause, preserves the two-party system
\fpause
Lots of our political disfunction can be blamed on the primacy of the two parties\pause, but most people see this as unavoidable.
\fpause
It's not. \pause It's caused by our use of the plurality system.
}

\frame{
Will Democratic and Republican politicians ever seriously consider dismantling the plurality system?
\fpause
The system which voters don't even think about, but the parties depend on for survival?
\fpause
%\only<1-3>{\phantom{ \[ \includegraphics[scale=1]{pigs.jpg} \] \hfill {\tiny{Picture from User:Durova at Wikimedia Commons, CC-BY-SA }}}}
\only<3>{ \[ \includegraphics[scale=1]{pigs.jpg} \] \hfill {\tiny{Picture from User:Durova at Wikimedia Commons, CC-BY-SA }}}
\only<4>{\[ \includegraphics[scale=.08]{face.JPG}\]}
\only<5>{\[\includegraphics[scale=.25]{monkey.jpg}\] \hfill {\tiny{Picture from Joel Telling at Flickr, CC-BY-SA}}}
}

\subsection{}
\frame{
The end!
\vfill
Read Wikipedia ``Voting system'' for lots more info and references.
\vfill
\url{http://faculty.fairfield.edu/cstaecker} for these slides
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