![](bg4.png)
the schedule {1.0, 0.8, 0.6, 0.4, 0.3, 0.2, 0.1}. Also as-
sume that d = 0.1 and δ = 0.7.
The buyer’s belief updates and decisions at every
stage are recorded in Table 1.
Both seller models may be used to rationalize the
first two offers. In these cases, Bayesian belief update
simply preserves the prior. The buyer rejects both
these offers because the discounted expected profit
from the next stage is greater. The next offer, 0.6, is
only rationalizable using Model 2. Therefore, the up-
dated belief concentrates the entire probability mass
on this model. The discounted expected profit from
the next stage is still greater – therefore, 0.6 too is
rejected. Finally, offer 0.4 is accepted.
4.2.1 Model Extinction and ACC
Example 2 usefully demonstrates the necessity and
sufficiency of the absolute continuity condition(ACC)
(Kalai and Lehrer, 1993) in an interactive decision-
making framework. Both models thought possible by
the buyer are wrong (in the sense that neither of them
represents the true seller). In fact, after the buyer re-
ceives offer 0.6, it (correctly) deems that the seller
cannot possibly be Model 1, and, therefore, removes
this model from the support of its beliefs. However, it
turns out that Model 2 accounts for the buyer’s obser-
vations all the way until the termination of the inter-
action (which happens when the buyer accepts offer
0.4). But note, in particular, that Model 2 is also a
wrong model of the true seller. Interestingly enough,
had the negotiations continued for one more round,
the next offer of 0.3 would have led the buyer to con-
clude that not even Model 2 could be the true model
of the seller, leading to the extinction of all models
in the buyer’s prior belief space. None of the prior
models it thought possible to begin with could suc-
cessfully explain (rationalize) reality; it realizes that
it was completely mistaken in its beliefs.
How should Bayesian decision-theoretic agents
prepare for such contingencies in multi-agent en-
vironments where it is uncertain about the type of
agent(s) with which it is interacting? As in our ex-
ample, consider an agent that starts with a prior belief
over a non-exhaustive set of possible models of the
opponent agent. If one of these models happens to be
the true model of the opponent, then our agent will
never be taken by surprise. In fact, after sufficient in-
teraction and observation, its beliefs will converge to
the true model (Kalai and Lehrer, 1993). If, on the
other hand, its beliefs do not contain the true model
in its support, then, barring a fortuitous satisfaction
of ACC, the agent will eventually be completely sur-
prised (an eventual extinction of all models in its be-
lief support).
Realization of ACC through Random Models. A
closer understanding of such an agent’s beliefs yields
a way out of this quandary. If an agent is so com-
pletely mistaken in its beliefs that the true model is not
even possible a priori, it is only natural that it even-
tually faces inexplicable situations. A more realistic
approach calls for a cautious agent that includes, in
the support of its beliefs, one more model – a random
model – which would make all actions (here, offer ev-
ery possible offer) plausible with some positive prob-
ability, and, thereby, account for all contingent behav-
ior (not already modeled by the other models). Such
a prior belief will always satisfy ACC. The following
example illustrates the usefulness of this approach.
Example 3. Consider a buyer, with valuation 0.7,
who believes, with probabilities 0.5, 0.4 and 0.1, re-
spectively, that the seller is one of three possible
types – a subintentional automaton with a fixed sched-
ule of offers, say {1.0, 0.9, 0.7, 0.4, 0.3, 0.2, 0.1}, or
an L0-Seller(d) type, or an L0-Seller(dr) type. Call
these Model 1, Model 2 and Model 3 respectively.
Suppose that the actual seller follows the schedule
{1.0, 0.9, 0.8, 0.7, 0.5, 0.3, 0.1}. As before, assume
that d = 0.1 and δ = 0.7.
The buyer’s belief updates and decision-making at
every stage are recorded in Table 2.
We observe that Model 1 is removed (from the
support of beliefs) when offer 0.8 is recieved. Model
2 is removed when 0.5 is recieved – leaving only the
random model, Model 3. Since this model rational-
izes every possible offer, the buyer is able to continue
interacting and eventually accepts 0.3.
5 L2-TYPE AGENTS
5.1 L2-buyer
The L2-Buyer “pre-solves” all its L1-Seller mental
models offline, incurring, in the process, an (offline)
polynomial time cost of
O
n
L1Seller
L2Buyer
×
|O|
2
× n
L0Buyer
L1Seller
Following this, its online operation is similar to that
of the L1-Buyer. Whenever an offer is recieved, it up-
dates its beliefs and decides whether or not to accept
by comparing the immediate profit with the expected
profit from the next stage – both of which are linear
time computations in the number of mental models –
namely, O
n
Seller
L2Buyer
.
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