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for straightforward expansion to the extensive form of this game. In this setting,
the agents are only interested in one item, and not even bundles of items.
Furthermore, we focus on experimental settings where the total capacity of
the agents exceeds demand, i.e. more loads can be transported than are available
for auction (
|
Agents
|∗
capacity >
|
Auctions
|
). This is in line with the field
of logistics where competition is intense and with competitive scenarios where
auctions are most appropriate. It forces the agents to formulate an aggressive
bidding strategy as items for auction are in high demand. In the design of the
bidding strategy, we do not have to explicitly consider the possible scenario of an
agent waiting until all other agents are full before trying to win auctions when
demand exceeds total capacity. In Section 4, we however do predict outcomes
for such scenarios.
We formulate a stochastic bidding strategy for an agent
a
. For each state
s
and
a successor state
s
(i.e. for one extra won load for one of the fruitful regions), we
define a local stochastic policy that chooses from a set of three bidding strategies
b
i
where
b
1
is the strategy of bidding the immediate valuation,
b
2
is the strategy
of overbidding the immediate valuation, and
b
3
is the strategy of bidding less
than the immediate valuation.
Strategy
b
1
acts as if there is no complementary valuation between the cur-
rent auction and future auctions. An agent using strategy
b
1
simply bids the
immediate valuation of a good as defined in Section 2. Strategy
b
2
has a more
aggressive line where a higher bid is submitted than the immediate valuation.
This is the bid as dictated by strategy
b
1
, but increased with a fixed, additive
bidmodifier
=0
.
1. Strategy
b
3
returns the bid as in strategy
b
1
but lowered by
bidmodifier
.Intuitively,strategy
b
1
is the “naive” or myopic bidder. Strategy
b
2
aims to acquire more than one item from the same fruitful region in order to
acquire the complementary benefit. Strategy
b
3
is also added to allow an agent
to back off from an auction. This reduce its chances of winning specific auctions
to allow an agent to reserve capacity for more profitable future loads
3
.Wedis-
cuss the impact of various settings of this bid modifier and the possibility for an
agent to learn to set this part of the strategy in Section 6.
For each state
s
and successor state
s
, a local strategy vector
sv
=
<p
1
,p
2
,p
3
>
where
Σ
i
p
i
= 1 is maintained where
p
i
is the probability of playing bidding
strategy
b
i
when entering an auction for a load
l
from state
s
. The policy, the
bidding strategy, of the agent is hence distributed over the state transitions
of the agents and is conditioned on the already won bids and expected future
possibilities. If
s
is however a state where full capacity has been reached, then the
fixed strategy vector is simply
<
1
,
0
,
0
>
as there are no future auctions to bid
strategically for. All other strategy vectors are initialised to
<
0
.
9
,
0
.
05
,
0
.
05
>
.
During one sequence of auctions (one epoch), for each agent
a
,aseparate
history
=
<h
1
,h
2
,...,h
n
>
is recorded where
h
i
registers the knowledge
a
has
of the results of the
ith
auction. This entails the results for bidding in the
ith
auction (loss or win and paid price), the state of
a
at that moment, and the
3
Experimental results (not shown) indicated that agents without the option of a
lowered bid performed significantly worse than their more versatile opponents.
