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\pagestyle{fancy} \lhead{ %Costandino Dufort
%Moraites
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\heading{1}{Spring 2012}{Instructor: Eva Tardos }{Costandino Dufort Moraites
(cdm82)
}{Lecture 1 Scribe Notes}

\section{Lecture 1 -- Monday 23 January 2012 - Mostly Intro}

\subsection{Overview} 

Special - possible extra credit for improving wikipedia articles on topics
related to this course.

Motivating connecting Game Theory and Computer Science. 

\begin{itemize}
\item The Internet. Computer Science used to be summarized by trying to make
computers effective. In comparison, today, there is the Internet connecting
everything and the idea that one person designing one algorithm controls
everything is just wrong. 

\item  Mechanism design. Really an engineering discipline, designing a game to
get a desired outcome. 
\end{itemize}

Different perspectives. Ways in which our perspective is different from the
traditional approaches to game theory. 

\begin{itemize}
\item We care about algorithms and as mentioned previously we know how to design
and analyse them. Economists come from a different historical perspective when
it comes to approaching these problems and there isn't the same emphasis on the
complexity of finding the equilibria of a system.  From the Computer Science
perspective, taking up this approach brings us to interesting complexity results
which we'll explore throughout the semester. 

\item Simplicity of mechanisms. Particularly coming from a systems background,
mechanisms must be simple. 

\item (Approximate) optimality. Say average response or delivery time.  Often
times, our objective functions don't have such clear cut measures of utility.
Time can be relative. Do we care about the difference between 7 and 8 seconds? 
\end{itemize}

See the web page for a rough syllabus for the course.

Since the book, mechanism design has changed a bit. We will be using Jason
Hartline's book on this topic more than the book listed as the text for this
course. 

Ken Binmoore basic introduction to game theory is a good, quick intro to game
theory. 

\subsection{An Example -- Braess' paradox}

We'll start by talking about Braess' paradox. Consider the following graph. 

\definecolor{qqqqff}{rgb}{0,0,1}

\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-5.0,-2.50) rectangle (7,3.7);
\draw [->,line width=2.8pt] (0.76,0.4) -- (3,2);
\draw [->,line width=2.8pt] (3,2) -- (5.42,0.34);
\draw [->,line width=2.8pt] (0.76,0.4) -- (3,-1);
\draw [->,line width=2.8pt] (3,-1) -- (5.42,0.34);
\draw (2.24,3.26) node[anchor=north west] {Strategy 2 };
\draw (2.58,-1.56) node[anchor=north west] {Strategy 1};
\draw (1.72,1.64) node[anchor=north west] {x};
\draw (1.92,0.18) node[anchor=north west] {100};
\draw (3.98,1.68) node[anchor=north west] {100};
\draw (4.02,0.18) node[anchor=north west] {x};
\begin{scriptsize}
\fill [color=qqqqff] (0.76,0.4) circle (3.5pt);
\draw[color=qqqqff] (0.9,0.82) node {$S$};
\fill [color=qqqqff] (5.42,0.34) circle (3.5pt);
\draw[color=qqqqff] (5.58,0.76) node {$T$};
\fill [color=qqqqff] (3,2) circle (3.5pt);
\fill [color=qqqqff] (3,-1) circle (3.5pt);
\fill [color=qqqqff] (-4.3,-8.62) circle (1.5pt);
\fill [color=qqqqff] (13.92,-8.62) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}

For every edge there is $d_e(x) =$ delay on $e$ if $x$ players use this edge. We
will assume $d_e(x)$ is a monotone increasing (i.e. non-decreasing) function.
Here strategy 1 is the bottom path and strategy 2 is the top path. 

Notice the two edges with weight 100. These edges take 100 seconds to cross
regardless of congestion, whereas the edges labelled $x$ are congestion
sensitive. 

Our goal: predict what will happen here. The obvious prediction is that the
players should split half-half. So 50 choose strategy one and another 50 choose
strategy two. This prediction is a \emph{(pure) Nash equilibirium} - an outcome
so that everyone chooses a strategy such that deviating from their strategy will
not improve their outcome. Total time is 150 for all players. If someone
following one strategy tries to deviate, they will not improve their outcome.
In this example, someone switching from strategy one to strategy two will
increase their travel time by a second worsening their outcome. 

Now we add an extra edge with no delay and see how this affects the strategies.
Adding the extra edge provides an additional strategy: strategy 3 where a player
starts out using the top edge, then uses the edge we just added and then takes
the last edge to the sink. 

Start with the 50-50 solution, but this is no longer an equilibrium. Now we
claim that all 100 choosing strategy 3 is a Nash equilibrium with total delay
200 for all players. 

\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.25cm,y=1.25cm]
\clip(-3.5,-1.00) rectangle (13.92,4.5);
\draw [->,line width=2.8pt] (0.8,2.24) -- (3.04,3.52);
\draw [->,line width=2.8pt] (3.04,3.52) -- (5.2,2.14);
\draw [->,line width=2.8pt] (0.8,2.24) -- (3.04,0.84);
\draw [->,line width=2.8pt] (3.04,0.84) -- (5.2,2.14);
\draw (2.46,4.3) node[anchor=north west] {Strategy 2 };
\draw (2.5,0.28) node[anchor=north west] {Strategy 1};
\draw (1.76,3.48) node[anchor=north west] {x};
\draw (1.96,2.02) node[anchor=north west] {100};
\draw (4.02,3.44) node[anchor=north west] {100};
\draw (3.78,1.9) node[anchor=north west] {x};
\draw [->,line width=2.8pt] (3.04,3.52) -- (3.04,0.84);
\draw (3.12,2.30) node[anchor=north west] {Strategy 3};
\begin{scriptsize}
\fill [color=qqqqff] (0.8,2.24) circle (3.5pt);
\fill [color=qqqqff] (5.2,2.14) circle (3.5pt);
\fill [color=qqqqff] (3.04,3.52) circle (3.5pt);
\fill [color=qqqqff] (3.04,0.84) circle (3.5pt);
\fill [color=qqqqff] (-4.3,-3.14) circle (1.5pt);
\fill [color=qqqqff] (7.52,-3.14) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}

To see this, notice that a player taking strategy 1 or strategy 2, then changes
his delay time from 200 to 200 so no improvement, thus we have the desired
equilibrium. 

This equilibrium is not unique. 99 could choose strategy 3 and 1 could
choose strategy 1. This would still be at equilibrium. The lonely guy might be
jealous of his 99 friends for getting delay 199, but changing his strategy
would not help his outcome from 200 so the system is still in equilibrium.
There are many more similar such equilibria. 

Claim that at least 98 players must choose strategy 3 and at most 1 player can
choose strategy 1 and at most 1 player can choose strategy 2 for Nash
equilibrium. 

Contrasting the dynamics of this game with the previous, is that now total
delay is 200 for most players and at best 198 for some players. In either case,
making life worse for all characters. Hence Braess' paradox -- adding an edge to
the graph made life worse for all the players. 

Questions we will look at: how do you find these equilibria? Why do we get
paradoxes like this one? Are situations like this paradox really so bad? Adding
the edge didn't make life that much worse for players. Can players find these
equilibria? 

We will look at various ways for looking at how to prevent paradoxes like this
one. We will heavily rely on some sort of `rationality' for the players. Later
we'll go back and look at a more intricate definition of equilibrium. Our
current definition is good enough when these equilibria are fairly stable and
unique. There are however criticisms of Nash Equilibrium and next time we'll get
some examples of where Nash Equilibrium doesn't capture everything we want. 

Next time we'll consider the significance of various objective functions. Maybe
you don't just want to make lots of money -- maybe it's more important to make
more money than your neighbors? Or maybe you don't just want to make as much
money as possible in a game, but rather to make sure you win the game. We will
explore trade offs between various measures of utility. 



\end{document}
