\section{Less competitive Payoff for learning}

%figures
\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{for-paper2.jpg}
\caption{SCG Binary Game Tree.}
\label{SCGextensive}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{SCGStructure.jpg}
\caption{SCG Structure.}
\label{SCGstructure}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{SCG-CSP.jpg}
\caption{Blame and Payoff Table}
\label{SCGCSPTable}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{SCG-CSP-Learning.jpg}
\caption{Blame and Payoff Table (Learning)}
\label{SCGCSPTableLearning}
\end{center}
\end{figure}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{blame-payoff-opt.jpg}
\caption{Blame and Payoff Table (Optimization)}
\label{SCGCSPTableOpt}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{GameTreeTable.jpg}
\caption{Blame and Payoff Table (with Game Tree)}
\label{GameTreeTable}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{GameTreeTableOpt.jpg}
\caption{Blame and Payoff Table (with Optimization)}
\label{GameTreeTableOpt}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{SCG-CSP-FSC-OPT.jpg}
\caption{Constraints Payoff Design (with Optimization)}
\label{SCGCSPFSCOPT}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{WS13-figures-agree.jpg}
\caption{Improved Agreement}
\label{ImprovedAgreement}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{WS13-Opt-justification.jpg}
\caption{Complete Table for Optimization}
\label{CompleteTable}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{WS13-all-blame.jpg}
\caption{All Learning Opportunities}
\label{AllLearning}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=85mm]{WS13-GTTOpt-master.jpg}
\caption{With Master Scholar}
\label{MasterScholar}
\end{center}
\end{figure}

Is this the correct translation from paper \cite{1}.

refute()? (pdsp,odsp):(parp,oarp)
agree()? (paso,oaso):(pdro,odro)

Depending on the application, many meaningful payoff functions can be defined.
For example, if SCG is used for creating student interaction in
a MOOC, I recommend the 
following low competition payoff function (see Figure 
\ref{SCGCSPTableLearning}. The values after / are for learning.
The competitive payoff function is shown before / ):


\begin{enumerate}
\item
refute:
\( p(c, ...)?(0,0):(0,1).\)
If the predicate is true, nobody gets a point because we want the Opponent
to learn from the Proponent through the refutation protocol.
If the predicate is false, the Opponent has won and gets a point.

\item
strengthen c to c':
\( p(c', ...)?(0,1):(0,0).\)
If the predicate is true, the Opponent gets rewarded with one point because he 
successfully defended the stronger claim.
If the predicate is false, the Proponent has won but does not get a point
because we want the Opponent to have cheap opportunities to attack and learn.

\item
agree:
\( p(c, ...)?(0,1):(0,0).\)
If the predicate is true, the Opponent has successfully defended the claim
and nobody gets a point. 
If the predicate is false, the Opponent has failed to defend the claim
but has gained information to learn. The Proponent earns a point.
\end{enumerate}
The motivation is that it should be penalty free for students to learn
from other students. Not succeeding in refuting a claim is free to the Opponent,
while successfully refuting gives a point to the Opponent.
Failing to strengthen a claim is free to the Opponent,
while successfully strengthening gives a point to the Opponent.
Failing to agree with a claim is free to the Opponent,
while successfully agreeing gives a point to the Opponent
This low competition payoff function has the flavor of a soccer game
where only the goals count.

The competitive payoff and the low competition payoff are two examples
of payoff functions that promote good behavior in the lab.
Other payoff functions are possible.

Instances are only available when they are needed.
For example, in the spirit of the Renaissance 
mathematical competitions between Tartaglia and Fior, if the protocol asks that the
Proponent  and Opponent deliver each 10 instances, followed
by the solution activity.
The instances are secret until they are solved.

\section{Game history}

Use it to measure learning.

Discuss learning from previous games.
In game G1 you disputed claim c and successfully refuted it.
In a future game G2 should you dispute it again?

Not necessarily, because you could fail to refute the
claim c in G2.

Reasons could be:

1. The proponent has improved and found a defense strategy for its claim c.

2. Although the claim is false, you lack a systematic
refutation strategy and in a second try you might fail to refute.

\section{Measure learning}

\subsection{Student Assessment with SCG}


SCG has an natural assessment approach implied by the Scientific Method.

\subsubsection{A perfect master teacher is available}

input: claim; output: true, false, optimal

input: true claim; output: instance that leads to defense

input: true claim, instance; output: does instance lead to defense?

input: true claim, instance; output: solution that defends claim

MAKE GENERIC

input: false claim. output: first step in refute(c,P,O) that leads to refutation.

input: false claim. Partial elaboration of refute(c,P,O) 
with next step to be made by O.
output: step by O that leads to refutation.

input: true claim. Partial elaboration of refute(c,P,O) 
with next step to be made by P.
output: step by P that leads to defense.

The above perfect master teacher capabilities can be used
to guide and assess the student.

\subsubsection{No perfect master teacher}

We still have the blame assigned based on the refutation 
protocol outcome (oB column in Figure \ref{SCGCSPFSCOPT}.

    reason for loss (e.g., proposed claim refutation) not easy to find
      claim could be false and properly attacked
        (error in propose)
      claim could be false and improperly attacked and improperly defended (error in propose,provide and solve)
      claim could be true but not properly defended
        (error in provide or solve)

don't know in which situation we are. How does SCG help?

Yes, SCG helps: reason: 
(oB column in Figure \ref{SCGCSPFSCOPT}.


\subsection{Learning Science and SCG}

I understand your concerns about incorporating learning scientists. I
believe, SCG has very good learning science built in. Below is a
description how learning happens and how it is measured in SCG.

In an SCG lab, learning happens during the elaboration of the
refutation protocol for a claim. When a claim is defended or refuted,
there is a sequence S of instances and solutions which has been
produced by the refutation protocol. If the claim is defended, the
claim predicate evaluates to true for S. The sequence S contains a
surprise for the opponent of the claim because the opponent's
intention was to make the predicate false.
This surprise is the crystallization point for learning. The student
playing the role of the opponent is encouraged to ask and answer the
following questions:
(O1) Why is my prediction wrong that I will successfully refute?
(O2) What is the general pattern behind the clever construction that
my partner used to defend the claim? Can I interfere with the clever
construction? Can I reconstruct it from S?
(O3) Can I defend the claim against a partner, successfully?
(O4) Can I improve my approach to trying to refute the claim in a
second attempt?
(O5) Do I still believe that I can refute the claim?
(O6) Did I make a  mistake? Was there a second or third mistake? Do a
blame assignment.

The proponent of the claim is pleased with winning but is not off the hook:
(P1) Did I win by accident? Has the opponent made a mistake which made
me win this time but not against a better partner?
(P2) How do I repeat my success even when the opponent plays differently?
(P3) Have I a systematic defense strategy?
(P4) Works my systematic defense strategy in all cases?

Emotions of the proponent when she wins: joy, I found a clever
construction to defend.
Emotions of the opponent when he loses: disappointment, I will try to
figure out your clever construction and maybe change my mind about
trying to refute.

SCG offers the following approach to measure learning in a lab for a
given student:
[unsuccessful => successful]
Defense attempts are unsuccessful (dau)=> defense attempts are
successful (das). Student learned to recognize, correctly, defensible
claims.
Refutation attempts are unsuccessful (rau) => refutation attempts are
successful (ras). Student learned to recognize, correctly, refutable
claims.
Agreement attempts are unsuccessful (aau) => agreement attempts are
successful (aas). Student learned to recognize, correctly, optimal
claims.
Amount learned: dau-das + rau-ras + aau-aas

[change of mind]
Claim C was unsuccessfully defended => claim C is successfully refuted
consistently
Claim C was unsuccessfully refuted => claim C is successfully defended
consistently
Amount learned: number of claims  where a change of mind happened.


\section{Small Labs}

Labs with c and !c.

\section{Labs with Perfect Avatars} 

Useful for learning. Always have perfect answers.
But costly to produce.

\section{Interesting Payoff Functions}

Looking at 
Figure \ref{SCGCSPFSCOPT}, there are two blame justifications
where O did not do anything wrong. It would be natural to give
a higher payoff to O in these two cases: odro = 2, osso = 2.

If O is blamed in oB, P might also have contributed misinformation: P might have proposed a false claim. It makes sense to
give a lower payoff to P: parp = 1, pdsp = 1, psrp = 1.
