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6 CONCLUSION
We propose and analyze a multi-item LUBA game
with budget constraint, registration fee and resubmis-
sion cost. We show that the analysis can be reduced
into a finite game (with incomplete information) by
eliminating the bids that are higher than the value of
the item or by the bid that are higher the total available
budget. Using classical fixed-point theorem, there is
at least one Bayes-Nash equilibrium in mixed strate-
gies. Next, we address the question of computation
and stability of such an equilibrium. We provide ex-
plicitly the equilibrium structure in simple cases. In
the general setting, we provide a learning algorithm
that is able to locate equilibria. We propose an im-
itative combined fully distributed payoff and strat-
egy learning (imitative CODI- PAS learning) that is
adapted to LUBA. We examine how the bidders of the
game are able to learn about the online system output
using their own-independent learning strategies and
own-independent valuation. The revenue of the auc-
tionner is explicitly derived in a situation where a ran-
dom number of bids are placed.
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PROOFS
Proof of Proposition 2. By budget constraint, j
0
s
bids must fulfill
∑
i
b
ji
≤
¯
b
j
. If b
i j
> v
ji
− ˜c, j gets
v
ji
− ˜c−b
ji
which is negative (loss), and j could guar-
antee zero by not participating. Therefore the strategy
0 dominates any b
ji
higher than v
ji
− ˜c.
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