Ahmed redesigned the game. It is nolonger about money but about 
reputation and confidence.

The goal of the game is to push the confidence of each challenge towards 1.
This is linked to maximizing the reputation of a player.

This leads to a better game.

Terminology:
belief
  problem flows from offerer to acceptor (acceptor solves) [FORWARD] or
                from acceptor to offerer (offerer solves)  [BACKWARD].

Example:
b1 = Belief classic ([(2,0) (1,0)] 0.65). 
Meaning of b1: Exists S
in niche so that for all J: fsat(S,J) < 0.65 and S can be found in polynomial
time. Belief of kind FORWARD.

b2 = Belief classic complement ([(2,0) (1,0)] 0.60). 
Meaning of b2: For all S
in niche there exists J: fsat(S,J) >= 0.60
and J can be found in polynomial time.
Belief is of kind BACKWARD.

b3 = Belief secret ([(2,0) (1,0)] 0.6). 
Meaning of b2: For niche Exists polynomial algorithm A
so that for all S in niche, the assignment J = A(S)
approximates the secret solution within 0.6, i.e.,
fsat(S,J) >= 0.6 * SQ.
Belief is of kind BACKWARD.

More terminology:
confidence (in a belief)
challenge = (belief,confidence)
agent
reputation (of an agent)
approximation factor (with respect to secret or another property of problem (like all constraints))

recognition (of a belief after the discounting protocol is done)
  a number in [-1,1].
Credibility (of a belief offered by agent) = 
  reputation (agent) * confidence (in belief)

A challenge consists now of a belief and a confidence in [0,1].
The confidence should represent the "true" confidence the agent
has in the belief. 
If the confidence is too high, the offerer loses 
more money when the challenge is discounted.
If the confidence is too low, the offerer makes less 
money when the challenge is supported.

(discounting a challenge means discounting the belief of the challenge)

If an agent has a constructive proof for the statement
behind a belief, the confidence should be set to one.

Ahmed's axioms:

1. Agents gain reputation either by discounting other agents' beliefs 
or by having their beliefs recognized by other agents. 
Agent's gain from their belief is proportional to both 
the credibility of their belief and the recognition of the belief
after the discounting protocol.
2. One agent's reputation gain is another 
agent's reputation loss. The sum of all agent's reputation is preserved.
3. Arguments (S and J) recognize beliefs by 
a recognition factor in [-1,1]. 
A recognition factor of 1 means that the other agent has 
completely failed to discount that belief. 
A recognition factor of -1 means that the other
agent has completely succeeded to discount that belief.
4. Beliefs have a credibility degree in [0..]. 
Credibility is proportional to either the agent's confidence, 
or the agent's reputation or both. 
The agent's confidence reflects the amount of effort 
made by the agent to prove the belief (i.e. turning it into a theorem). 
Agent's reputation is the accumulation of the agent's initial reputation 
and its reputation gains and losses; thus reflecting the past performance of the agent.