A Graphial method for simplifying Bayesian
games
Peter Thwaites and Jim Q. Smith
Department of Statistis
University of Warwik
Coventry, United Kingdom
May 2011
If the inuene diagram (ID) depiting a Bayesian game
is ommon knowledge to its players then additional assumptions may
allow the players to make use of its embodied irrelevane statements.
They an then use these to disover a simpler game whih still embodies both their optimal deision poliies. However the impat of
this result has been rather limited beause many ommon Bayesian
games do not exhibit suÆient symmetry to be fully and eÆiently
represented by an ID. The tree-based hain event graph (CEG) has
been developed speially for suh asymmetri problems, and new
separation theorems for these graphs make it possible for rational
players to make analogous dedutions, assuming the topology of the
CEG as ommon knowledge. In this paper we desribe these powerful new tehniques and illustrate them through an example modelling
risks to eletroni ommuniation.
Abstrat:
Bayesian game theory, hain event graph, deision tree,
inuene diagram, parsimony
Keywords:
1
Introdution
There are two prinipal oneptual diÆulties in applying results from Bayesian
game theory in a number of domains. Firstly, whilst it might be plausible for a
player to know the broad struture of an opponent's utility funtion when that
opponent is subjetive expeted utility maximizing (SEUM), for a player to also
believe that she knows the exat quantitative form of that utility funtion or the
preise formulation of the distribution of its attributes strethes redibility. Seondly, as for example Nau [15℄ has pointed out, however ompelling our beliefs
are that an opponent's rationality should indue her to be SEUM, in pratie
most people simply are not. So any appliation of a theory whih starts with
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this assumption is liable to fall at the rst hurdle. These issues indued Kadane
and Larkey [12℄ to suggest giving up on the rationality hypothesis entirely and
instead modelling the opponent simply in terms of her past behaviour.
However others have perservered with rationality modelling by addressing
these real modelling hallenges more qualitatively. For example Smith [25℄ suggested a way to address the rst diÆulty desribed above. We need the onditional independenes assoiated with various hypotheses and the attributes of
eah player's utility funtion to be ommon knowledge, but we do not need that
the players know the quantitative forms of others' inputs. This framework developed from methods for simplifying inuene diagrams (IDs) [8, 10, 29℄, desribed
rst in [20℄ and then [24℄. When players are all SEUM substantive onlusions
an sometimes be made onerning those aspets of the problem upon whih a
rational opponent's deision rules might depend. This in turn allows players to
determine ever simpler forms for their own optimal deision rules. This enables
models to be built whih at least respet some of the strutural impliations of
rationality hypotheses before being embellished with further struture gleaned
from behavioural data, or the bold assumption that an opponent's quantitative
preferenes and beliefs an be fully quantied by everyone. Even the seond
ritiism of a Bayesian approah outlined above is at least partially addressed
sine the methods need only ertain strutural impliations of SEUM to be
valid, not that all players are SEUM. In this way formal game theory is used
not to fully speify the quantitative form of a ompetitive domain but simply
to provide hypotheses about the likely dependene struture that rationality assumptions might imply for suh models. These models an then be embellished
with further historial quantitative information using the onventional Bayesian
paradigm.
It has been possible to demonstrate the eÆay of the approah when modelling ertain rather domain-spei appliations, but it has proved rather limited in sope. One problem is that the struture of many games an not be
fully and eetively represented by an ID. Usually the underlying game tree is
highly asymmetri and so the symmetries neessary for an enompassing and
parsimonious ID representation of the game are not present.
We have reently developed a powerful new graph alled a hain event graph
(CEG) partiularly for the modelling of problems whose underlying trees exhibit
a high degree of asymmetry. This provides a platform from whih to dedue
dependene relationships assoiated with variables dened by the graph diretly
from its topology [27, 31℄. These results have now been extended to provide
algorithms enabling us to disover from the topology and olouring of the graph
minimal sets of variables needed to fully speify an SEUM deision rule [6℄.
In this paper we demonstrate for the rst time how these new qualitative
strutures an be used to rene a nite game tree and so disover simpler games
whose optimal deision rules for two SEUM players are also optimal deisions of
the original problem. This is based on the assumptions that a partiular CEG
is ausal and its topology is ommon knowlege. In the next setion we briey
introdue the semantis of the CEG and the idea of parsimony as it applies to
this new graph. We then proeed to illustrate the use of these tehniques and
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reformulate a game tree whih models attempts to disrupt a ommuniation
system.
2
Chain event graphs, ausality and parsimony
The essential CEG is a funtion of an event tree [21℄ whose topology expresses
the onditional independene struture of a problem [31℄. Under ertain assumptions Bayesian networks (BNs) an be onsidered ausal, and given analogous
assumptions CEGs an also be onsidered as ausal [32, 30℄. Heuristially this
means that the model speied by a CEG ontinues to be valid when partiular variables are manipulated (see for example [16℄ for ausal manipulation of
BNs). Note that suh a hypothesis is a partiularly natural one to entertain in a
game whose underlying struture is ommon knowledge and where eah player
is able to manipulate their own deisions to a partiular value, but nature or
the player's opponents will determine the value of other variables.
If we allow more than one node to be manipulated on possible paths through
the CEG, then these nodes an be onstrued as deision nodes and the CEG
as a funtion of a deision tree [19, 14, 28℄. The remaining nodes in the CEG
then beome hane (or utility) nodes. In this mode the CEG is an elegant
answer to the problems highlighted in [4, 23, 17, 3, 2℄ where dierent ations
an result in dierent hoies in the future. As suh it provides an alternative
to the valuation networks of Shenoy [22℄, deision iruits of Bhattaharjya and
Shahter [1℄ and the sequential IDs of Jensen et al [11℄.
In [32℄ and [30℄ we were onerned primarily with the eets of a manipulation and whether these eets ould be gauged from probabilities in the idle
system. Considering the CEG as a funtion of a deision tree in ontrast, we
assume that the owner of the CEG has a utility funtion over the possible outomes of the problem, and an then use tehniques analogous to those used for
deision trees to nd an optimal deision rule for the deision maker [6℄.
Figures 1 and 2 illustrate how a deision tree is onverted into a CEG.
Figure 1 shows a variation [7℄ of the familiar Oil drilling example from [18℄.
We have an option on testing some ground for oil. We an either
take up this option, at a ost of 10, or pass it on to another prospetor
for a fee of 20. Whoever does the testing, the outomes are good or
bad, with probabilities independent of who does the testing. If we
have passed on the testing and the test result is good then we lose
the option for drilling and get nothing. If it is bad then the other
prospetor will not drill and the option for drilling reverts to us.
If we do the test and the result is good, then we an either drill,
for a ost of 30, or sell the drilling rights for a fee of 40. If the
result is bad, then regardless of who does the test, we an either
drill ourselves, again for a ost of 30, or sell the drilling option for
a fee of 10. If we drill and nd oil we gain 100.
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0
good
10
100
no drill 0
oil
pass
+20
bad
drill -30
no drill 0
good
test
-10
0
no oil
40
100
oil
drill -30
no oil
no drill 0
0
10
bad
oil
100
drill -30
no oil
0
Figure 1: Coloured deision tree for Oil drilling example
Deision nodes are indiated by squares, hane nodes by irles and utility
nodes by diamonds. Utilities are deomposed and appear both on the leaf
utility nodes and on the edges emanating from deision nodes. These edges are
oloured if they represent the same deision and arry the same utility. Edges
emanating from hane nodes are oloured if they represent the same outome
and arry the same probability. The atual probabilities are reported in [7℄.
This olouring of the deision tree enodes the onditional independene
struture of the problem, and we use it for two denitions. Nodes whih have
idential olouring on their emanating edges are said to be in the same stage.
So both the good /bad hane nodes are in the same stage, as are all the drill /no
drill deision nodes. Nodes whih root subtrees whih are idential in both
topology and olouring are said to be in the same position, so for example the
rst and third drill /no drill deision nodes are in the same position.
In the CEG nodes in the same position are ombined into a single vertex, and
the olouring is retained from the tree. The utility nodes are onneted to a terminal super-utility node (as exists in some ID representations, for example [5℄).
The CEG orresponding to Figure 1 is given in Figure 2.
In the example above we have a single deision maker (DM). When we move
into game theory the CEG represents the problem to eah of the players and its
topology an be onsidered as ommon knowledge (CK). The underlying struture is ausal to eah player, but eah deision node an only be manipulated
by a single (speied) player. In [25℄ we onsidered IDs whih obeyed the same
assumptions as those desribed immediately above. Suh IDs were resurreted
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good
no drill
40
pass
+20
bad
oil
good
drill
-30
100
no oil
test
-10
oil
bad
0
drill -30
no oil
10
no drill
Figure 2: CEG for Oil drilling example
and modied by Koller and Milh [13℄ and alled MAIDs (multi-agent inuene
diagrams). In [24℄ we noted that suh IDs ould be seen from the point of view
of an informed observer to whom all nodes ould be onsidered as hane nodes
- a senario whih for example would be valid if the observer's BN were ausal
and ommon knowledge. In [25℄, building on work in [26℄, we also noted that
when a player needs to make a deision at D, some of the information they
have obtained beforehand may be superuous for the purposes of making this
deision. Then we an write
U q QS (D) j (D; QR (D))
(1)
where U is the utility for that player, and Q(D) = fQS (D); QR (D)g
(S = superuous, R = required) is a partition of the information obtained
by the player before making the deision (equivalent to the verties in the ID
with edges direted into D). In this ase we need only onsider the onguration of the variables QR (D) when hoosing a deision at D to maximise U . We
all this priniple parsimony. In [13℄ the authors use a similar idea they all
strategi relevane and onnet this to a onept alled s-reahability.
It is shown in [31℄ that there are diretly analogous results for CEGs to
d-separation in BNs, and so (as implied earlier) the onditional independene
struture of a problem an be read from its CEG. The idea of the informed
observer allows us to extend the results to IDs and MAIDs. We demonstrate
this in the next setion, where we also show how the idea of parsimony allows
us to onsiderably simplify the analysis of multi-player games.
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3
Example
The ID of this example (Figure 3) appeared originally in [25℄ where it was
used to demonstrate how the parsimony assumption an simplify the analysis
required in a 2 person game. The problem is presented here in the form of a
CEG (Figure 4) within the ontext of eletroni ommuniation and message
seurity. We demonstrate that this same idea of parsimony an be used to
simplify CEG-based analysis, but we also show how CEG methods aomodate
problem asymmetry in a way that unmodied IDs do not. Moreover we see how
the proess of simpliation an be diretly linked to the asymmetries exhibited
in the assoiated game tree. Note that it is plausible that the CEG in Figure 4
is ausal and ommon knowledge.
In our example, one player A is in ommuniation with a third party C ,
and the other player B is attempting to deode the messages sent by A. The
ID and DEG ontain hane and deision nodes as desribed below. A joint
utility funtion for A and B is given as a funtion of the outomes of the hane
variables X3 and X4 .
D2(A)
D1(A)
X2
X1
X4
D1(B)
D2(B)
X3
U
Figure 3: Initial ID
X1 : A message is sent from C to A via one of two routes. This information
is seure (ie. it is not CK), but B observes whih of the two routes is
sent by.
D1 (A): A replies in ode to C (message ) using either ode I or II. B observes
whih ode A uses, but without a key has no guarantee of raking it.
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X3
X2
D1(B)
D1(A)
U
D2(A)
32 nodes
X1
X4
64 nodes
D2(B)
Figure 4: CEG of model depited in Figure 3
D1 (B ): B uses either tehnique Y or Z to try and deode message . Whih
tehnique B uses is observed by A.
X2 : The fat that B attempts to deode message may send a warning to C .
The probability of it doing so depends on whih of the two routes was
sent by, whih ode A used for , and whih tehnique B uses to try and
deode . Whether or not this warning is sent is observed by both A
and B .
X3 : Either B fails to deode with utility (UA3 ; UB3 ) = (+10; 0), or sueeds
with utility (UA3 ; UB3 ) = ( 10; +10). B 's suess depends on whih ode
A used and on whih tehnique B used. B 's suess or otherwise is observed by A.
D2 (A): A sends a seond message to C . The information ontained in this
message is false, but this fat is only known to A and C (ie. it is not CK).
A uses either ode III or IV. B observes whih ode A uses, but without
a key has no guarantee of raking it.
D2 (B ): B uses either tehnique Y or Z to try and deode message .
X4 : Either B fails to deode with utility (UA4 ; UB4 ) = (0; 0), or sueeds.
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Table 1: (UA ; UB ) as a funtion of X3 and X4
X4
B failure
X3
B suess
B failure (+10, 0) (+30, +10)
B suess (-10, +10) (-10, +10)
The CEG is designed for use with asymmetri problems. ID-representations
often obsure suh asymmetries and so even if a problem is expressed as an ID it
might still inorporate signiant asymmetries. So in this example, B might only
learn what deision A makes at D1 (A) if X1 takes the value 2 (the message is
sent via the seond route); X2 might only depend on X1 if A makes the deision
to use ode I at D1 (A) and so on. Clearly our DEG an depit expliitly any
number of asymmetries like this, but for illustrative onveniene we onentrate
here on just one suh possibility.
If B failed to deode but sueeds in deoding then he believes
the information ontained in the latter message with utility (UA4 ; UB4 ) =
(+20; +10). If B sueeded in deoding he does not believe the information in with utility (UA4 ; UB4 ) = (0; 0). B 's suess depends on whih
ode A used and on whih tehnique B used.
Note that UB4 (X3 = 1; X4 = 2) (B fails to deode but sueeds in deoding )
is positive beause B believes the information ontained in .
UA4 (X3 = 1; X4 = 2) > UB4 (X3 = 1; X4 = 2) as A has suessfully planted
false information on B .
Note that if B sueeds in deoding then the deisions made at D2 (A); D2 (B )
have no inuene on U , and neither does the outome of X4 . The ID in Figure 3 is still a valid representation of this problem, but we an inorporate this
information into the CEG from Figure 4 as desribed below.
For illustrative onveniene the CEG in Figure 4 has been drawn out in more
detail than is stritly neessary. In fat it is only neessary for the diagram to
show ertain key aspets of the model { the olouring of the verties assoiated
with X3 ; the grouping and olouring of the X4 verties; and the asymmetri
grouping of the utility verties. These four aspets orrespond to the four nontrivial onditional independene statements assoiated with the model.
In the CEG the X3 verties are grouped into four stages (indiated by the
olouring of the nodes), reeting the fat that the outome of X3 does not
depend on the outome of X2 nor diretly on the outome of X1 . Similarly the
X4 verties are also grouped into four stages, reeting the fat that the outome
of X4 depends diretly only on the deisions made at D2 (A) and D2 (B ), both of
whih deision nodes are binary. There are eight X4 verties (as opposed to the
128 in the underlying tree) beause the utility funtion depends on the value
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taken by X4 but also on the value taken by the binary variable X3 (whih has
no diret inuene on X4 ).
X2
D1(B)
X3
D2(A)
D2(B)
X4
U
D1(A)
X1
Figure 5: First redued CEG
We use the notation X3 ?G (X1 ; X2 ) j (D1 (A); D1 (B )) to indiate that our
ability to predit the value of X3 need not be inuened by the values of X1
and X2 if we know the deisions made at D1 (A) and D1 (B ). This formalizes
the desription of the X3 stages given earlier. The notation ?G stresses that
this is a property readable from the topology of the graph.
Similarly we an read the X4 stages to give us
X4 ?G (X1 ; D1 (A); D1 (B ); X2 ; X3 ) j (D2 (A); D2 (B ))
But sine U is also dependent on X3 , reading the X4 positions gives
(X4 ; U )?G (X1 ; D1 (A); D1 (B ); X2 ) j (X3 ; D2 (A); D2 (B ))
(2)
Reading the penultimate utility nodes gives
U ?G (X1 ; D1 (A); D1 (B ); X2 ; D2 (A); D2 (B )) j (X3 ; X4)
We now turn our attention to how parsimony allows us to simplify analysis.
As already noted, for a player making a deision at D if she an learn that
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D2(B)
D1(B)
X3
X4
D2(A)
U
D1(A)
Figure 6: Parsimonious CEG
U q QS (D) j (D; QR (D)), then she need only onsider the onguration of the
variables QR (D) when hoosing a deision at D to maximise U . Now, if in our
CEG representation she learns for example that
(Xe ; Df ; U )?G (Xa ; Db ) j (X ; Dd )
then it is shown in [6℄ that in an obvious sense
(Xe ; U ) q (Xa ; Db ) j (X ; Dd )
So our four important topologially derived properties an all be rewritten as
onditional independene statements.
Note that in Figure 4, D2 (B ) depends on all the preeding variables (in the
fully drawn out CEG there are 64 unoloured D2 (B ) verties), so Q(D2 (B )) =
fX1; D1 (A); D1 (B ); X2 ; X3; D2 (A)g. But expression (2) gives us that
U q (X1 ; D1 (A); D1 (B ); X2 ) j (D2 (B ); X3 ; D2 (A))
and so QS (D2 (B )) = fX1 ; D1 (A); D1 (B ); X2 g. These variables an be onsidered as non-parents of D2 (B ) for the purposes of optimal deision making and
we an ombine the 64 D2 (B ) verties into four, eah with 16 inoming edges
orresponding to the 16 possible ongurations of (X1 ; D1 (A); D1 (B ); X2 ). Eah
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X4 vertex now has only 1 inoming edge. Also as all X3 = 2 edges lie on sub-
paths leading to the same penultimate utility node we an simply replae eah
of these subpaths with an edge from an X3 vertex to this utility node.
We now repeat this proess for the verties D2 (A); D1 (B ) and D1 (A) using the new onditional independene properties depited in the most reently
pruned graph together with expression (1). After the rst two iterations we get
the CEG in Figure 5, and after a few further iterations we get the parsimonious
CEG in Figure 6. We an read from this graph that X1 and X2 are irrelevant for
optimal deision making purposes; and X3 depends on both D1 (A) and D1 (B ).
If X3 = 2, deisions made by A and B at D2 (A), D2 (B ) are irrelevant. Only
if X3 = 1 are they of any onsequene and A need not onsider anything else
when making the deision D2 (A). In this ase B needs only to onsider D2 (A)
when making the deision D2 (B ). U depends only on the values taken by X3
and X4 .
The optimal deision rules for A and B are found simultaneously by treating
the CEG from Figure 6 as if there were a single deision maker, working upstream from the terminal node, but noting that when we reah the set of D2 (B )
verties, the optimal deisions will be those that maximise B 's utility, and that
when we reah the set of D2 (A) verties, the optimal deisions will be those
that maximise A's utility and so on.
4
Disussion
We hope that the very simple example above demonstrates how Bayesian game
theory an be partiularly useful in onstruting models of ompetitive environments based only on the impat the likely rationality of players might have on
the struture of the game. Having identied a lass of models onsistent with
the derived strutural hypothesis a modeller an proeed to use more standard
Bayesian methodologies to estimate its parameters. Bayesian game theory an
thus be used to enhane and omplement a Bayesian analysis, making it more
plausible from a perspetive of mutual rationality. The reent developments in
tree models [9, 21, 4, 1, 11, 27, 32℄ makes this a partiularly powerful modeling
tool for environments with adversarial deision making. We will report these
results more fully in a later paper.
Aknowledgements
This researh is supported by the EPSRC, grant no. EP/F036752/1.
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