Information Technology Reference
In-Depth Information
(D,D), then once again both loads are won by one agent. A loss in profit of
is
however incurred due to the aggressive bidding of the opponent.
According to the classic definitions of the PD, the payoff matrix must meet
the strict conditions
T>R>P>S
and
R>
S
+
2
. According to [13], it is
however a sucient condition for one player to fulfill the condition that ((0
.
25
−
)
0
.
25)
>
0 for this player to have as dominant strategy to play
defect. As the payoff matrix is symmetrical, both players will converge to the
(D,D) equilibrium.
More generally, the
n
-player Prisoners' Dilemma game can be defined as in
[17]:
−
0)(0
.
5
−
1. each player faces two choices between cooperation (C) and defection (D);
2. the D option is dominant for each player, i.e. each has a better payoff choosing
D than C no matter how many of the other players choose C;
3. the dominant D strategies intersect in a deficient equilibrium. In particular,
the outcome if all players choose their non-dominant C-strategies is prefer-
able from every player's point of view to the one in which everyone chooses
D, but no one is motivated to deviate unilaterally from D.
More formally, the conditions used in [17] are: (1)
D
i
>C
i
for 0
≤
i
≤
n
−
1;
(
D
i
+
C
i−
1
)
2
(2)
D
i
+1
>D
i
and
C
i
+1
>C
i
for 0
≤
i<n
−
1; (3)
C
i
>
for
0
1. The payoff matrix is symmetric for each player. Here
C
i
denotes
the reward for cooperating with
C
i
cooperators and
D
i
the reward for defecting
with
i
cooperators and
n
≤
i
≤
n
−
1 other defectors
5
.
In Figure 4a we give the (average) payoff of one agent playing with 10 agents
for 3 fruitful regions of value
<
9
,
9
,
9
>
as given by experimental results. We
show the expected payoffs for selecting a cooperative (myopic bidding) or a defect
strategy (an overbidding strategy) as function of the total number of cooperators
in the game.
Analysis of a 2 player situation with a payoff matrix of the form of Table 4b
with the rest of the agents invariant and of which 0
−
i
−
8choosetocooperate,
shows that each individual agent, when deliberating in isolation, will converge
to defect. The conditions of [13] are again met. Furthermore, [17] showed in
computational experiments that coalitions of cooperators with 8 or more players
were extremely dicult to realize. In [1], these results are improved by allowing
tagging of individuals to enable agents to track defectors, but still cooperation
is extremely tenuous. The auction mechanism as currently defined in Section 2
anonymises the individual agents and precludes tagging. Lastly, the payoffs for
a choice of defect in Figure 4a greatly exceed the bounds of the third condition,
C
i
>
(
D
i
+
C
i−
1
)
≤
n
≤
2
,ofthe
nIP D
as used in the above work, leading to a stronger
preference for a defect strategy by the agents.
For new domains, and novel settings, it is worthwile to compare the perfor-
mance of a simple bidding strategy, like the one in Section 3, to a myopic bidder
5
Note that these constraints do not reduce to the classic PD for two players, but
thankfully do meet the weaker constraints derived from [13]. The used payoff table
for the experiments of [17] however do meet the 3 criteria given.
