Eco514|Game Theory
Lecture 4,Games with Payo Uncertainty (1)
Marciano Siniscalchi
September 28,1999
Introduction
The vast majority of games of interest in economics,nance,political economy etc,involve
some form of payo uncertainty,A simple but interesting example is provided by auctions:
an object is o ered for sale,and individuals are required to submit their bids in sealed
envelopes,The object is then allocated to the highest bidder at a price which depends on
every bid,according to some prespeci ed rule (e.g,a \ rst-price" or \second-price" rule),In
many circumstances (e.g,mineral rights auctions) it is reasonable to assume that the value
of the object is not known to the buyers,but that they receive some signal correlated with
it,In other cases,each buyer knows the value of the object to her,but not the other players’
valuation.
We can model this situation assuming that the set of actions available to every player
is the nonnegative real half-line,or some subset of it,representing allowable bids,While a
pro le of bids speci es the \outcome" of the auction,i.e,who receives the object and what
price she pays for it,it does not specify the payo to the winning bidder.
Her payo is determined by the (uncertain) value of the object; the uncertainty may even
be resolved only after the bidding game is over (e.g,when drilling or mining actually begins,
in the \mineral rights" case).
Thus,we need to extend our basic model of simultaneous games in order to account for
payo uncertainty.
The second issue is related to the solution concept(s) to apply in this setting,In the
auction example,it is reasonable to assume that players will condition their bid on whatever
signal they receive,In particular,in the \mineral rights" case,for any pro le of bids,players
cannot compute their actual payo because of the underlying uncertainty,but they can at
least compute their conditional expected payo given their signal.
This implies that a solution concept for games with payo uncertainty should specify an
action (or a set of actions) for each player and for each realization of whatever signal that
1
player may observe.
However,several subtle issues arise regarding the interpretation of \signals." If the
latter can be literally taken to be physical observations (e.g,samples or geological surveys),
then it makes sense to assume that players share a common probabilistic description of
the underlying uncertain quantities and their correlation with each player’s signal,Such
description could even be said to be part of the formulation of the game.
If,however,signals do not have a readily available physical interpretation (consider the
value,i.e,the maximum amount one is willing to pay for a painting,which presumably
includes a subjective element),then it does not make much sense to postulate that players
agree on any given probabilistic model of the underlying uncertain quantities|let alone make
such description part of the formulation of the game,Rather,a speci cation of players’ beliefs
about the underlying uncertainties should be considered as part of the solution concept.
Further observations will be in order once we specify a formal model of games with payo
uncertainty.
Games with Payo Uncertainty
I follow OR (with minimal expository deviations) and begin by providing a rather general
de nition.
De nition 1 A ( nite) normal-form game with payo uncertainty is a tupleG = (N; ;(Ai;ui;Ti)i2N),
where N is a nite set of players,is a set of states of the world,and for each i2N,Ai is
a set of actions,ui,Ai A i !R is Player i’s payo function,and Ti is a partition of
,referred to as Player i’s type partition.
It may be helpful to relate normal-form games with payo uncertainty with the decision-
theoretic framework introduced in Lecture 1,Taking the perspective of Player i,the set of
states of nature for i is ~ = A i ; the set of acts is (isomorphic to) Ai,with each act
ai 2Ai mapping a state (a i;!) 2 ~ to the outcome determined by the tuple (ai;a i;!);
nally,Player i’s preferences over outcomes are represented by ui.
Thus,Player i is not only uncertain about her opponents’ actions,she is also uncertain as
to the prevailing state of the world !2,In keeping with our general framework,a solution
concept should specify a probability distribution over whatever a player is uncertain about;
hence,it must also specify her beliefs about the prevailing state of the world.
Note,the usual \textbook" presentation of the situations referred to herein uses the
terminology \games with incomplete information,and includes a speci cation of a common
prior over the set of states of the world as part of the description of the strategic situation.
We use the somewhat nonstandard term \payo uncertainty" to emphasize that we do not
wish to regard probabilities over states of the world as part of the model,but rather as part
2
of the solution to the model,In particular,we do not wish to assume that such probabilities
are commonly agreed upon among the players,Notice that OR follows the same approach.
The key elements are the set of states of the world,which also a ect payo s (otherwise
there could be no payo uncertainty!),and the players’ type partitions.
The idea is that,if the prevailing state is !2,each player i2N is only noti ed that
the prevailing state is in the unique cell ti2Ti such that !2ti; that is,Player i rules out
any !062ti,but regards any !02ti as possible,We denote by ti(!) the set of states that
Player i deems possible at !2,that is,ti(!) is the unique ti2Ti such that !2Ti.
Indeed,in order to interpret the partition Ti correctly,one should think of states in each
cell ti2Ti as virtually indistinguishable from the point of view of Player i,This will be clear
from our use of the model.
To x ideas,consider the following version of the \mineral rights" auction setting,there
are two bidders,i = 1;2,and each receives a signal si2f0;1;:::;10g; the value of the object
(concession) is v(s1;s2) = s1 +s2.
Thus,=f0;1;:::;10g f0;1;:::;10g; Ti =ff(s1;s2),si = kg,k = 0;1;:::;10g; and
ui(ai;a i;(s1;s2)) = s1 +s2 ai if ai >a i,12(s1 +s2 ai) if ai = a i,and 0 otherwise.
The key observation is that each bidder i2N only observes her \part" of the true value,
si,On the other hand,both her signal and her opponent’s signal determine payo s.
Elements of the partition Ti are also referred to as types,Each type arguably captures a
possible information state of Player i,it tells us all she knows about the prevailing state of
the world.
Note well that,on the other hand,a type generally does not uniquely determine a player’s
payo corresponding to each action pro le,this is illustrated in the mineral rights auction
example above,Perhaps more interestingly,one could think of di erent information states,
sayti;t0i2Ti,such that,for all!;!02ti[t0i,ui(ai;a i;!) = ui(ai;a i;!0) for all (ai;a i)2A:
for example,this might be the case if,at ! and !0,player i attaches the same value to a
painting,but holds di erent beliefs as to the prevailing state of the world (hence,about her
opponents’ valuation),More on this point later.
Bayesian Nash Equilibrium
The standard analysis of normal-form games with payo uncertainty (suitably extended to
accommodate our decision-theoretic view) is centered upon the notion of Bayesian Nash
Equilibrium,due to John Harsanyi:
De nition 2 A Bayesian Nash Equilibrium of a normal-form game with payo uncertainty
G = (N; ;(Ai;ui;Ti)i2N) is a tuple (pi;( i;ti)ti2Ti)i2N such that,for all i2N:
(1) pi2 ( ) and,for all ti2Ti,pi(ti) > 0;
3
(2) For all ti2Ti and for all a0i2Ai,
X
!2ti
pi(!)
pi(ti)
8<
:
X
a i2A i
Y
j6=i
j;tj(!)(aj)
!" X
ai2Ai
i;ti(ai)ui(ai;a i;!)
!
ui(a0i;a i;!)
#9=; 0
Note that it is of course su cient to check for pro table deviations to pure actions.1
The product in parentheses is the probability that the action pro le a i will be played
in state !2,This is the product of the probability that each aj,j6= i,gets played,given
that,in state !,player j6= i is of type tj(!).
Next,the expression in curly braces is the di erence in expected payo from playing ai
and a0i in state !.
Finally,these di erences are weighted by the posterior probability that the state is actu-
ally !,given that the true state must lie in ti.
Observe that,it is also possible to formulate the best-reply property for type-contingent
randomized actions i = ( i;ti)i2N,As long as condition (1) above holds,it can be shown
that the resulting de nition is actually equivalent to the one given above,I will ask you to
provide the details.
Where does this come from?
We now take a step back and try to provide a rationale for this seemingly complex construc-
tion (due largely to John Harsanyi,a recent Nobel laureate),It turns out that,indeed,this
framework greatly simpli es the problem of describing strategic interaction in games with
payo uncertainty.
To keep the exposition simple (and in view of the fact that auctions will be the subject
of a forthcoming lecture),let us consider a simple two- rm Cournot model with uncertainty
about Firm 1’s cost.
Denote the latter variable by c2f0; 12g,Recall the setup,demand is given by P(Q) =
2 Q and each rm can produce qi2[0;1],Firm 2’s cost is zero.
It is probably easiest to think in standard equilibrium terms,Suppose that we are told
that a given tuple (q1;0;q1;1
2;q2) is \an equilibrium outcome" of the game,Let us ask ourselves
exactly what this implies.
At rst blush,there is little action on Firm 1’s side,she (?) expects Firm 2 to produce
q2 units,as per the equilibrium prescription,and best-responds to this,q1;c = BR1(q2;c) =
(1 c2) 12q2,for c2f0; 12g.
Now let us concentrate on Firm 2,Clearly,the latter cannot compute a best reply to
his opponent’s actions without formulating a conjecture about the relative likelihood of the
1Also note that,strictly speaking,what is de ned above is Bayesian Nash equilibrium for the mixed
extension of the original game G.
4
events c = 0 and c = 12,Denote by the probability of the former event; then,since Firm
2’s best reply is BR2( q1;0 + (1 )q1;1
2
),if we are given the information that q2 is a best
reply,we can \reverse-engineer" the value of (in this case,it is unique).
In terms of our model,we could let =f0; 12g,T1 =ff0g;f12gg,T2 =f g and p2(0) =
,It is easy to see that specifying p1 is not relevant for the purposes of De nition 2:
what matters there are the beliefs conditional on each t1 2T1,but these will obviously be
degenerate.
Let us think about the value of the parameter (or,equivalently,of the probability
p2),We have just noted that,given the equilibrium outcome,it is possible to derive Firm
2’s conjecture,Hence,since Firm 1 expects Firm 2 to play q2 and (according to the
equilibrium assumption) believes that Firm 2 expects Firm 1 to play q1;c,for each value
of the cost parameter c,it follows that Firm 1 can infer that Firm 2’s assessment of the
probability that c = 0 is,In other words,implicit in the speci c equilibrium we are looking
at is an assumption about Firm 1’s beliefs regarding Firm 2’s beliefs,It is easy to see that
we can continue in this fashion to build a whole hierarchy of interactive beliefs.
In terms of our model,note that Player 2’s type partition is degenerate,As a consequence,
at any state ! 2,Player 2’s conditional beliefs about,pi( jt2(!)) are the same|they
are given by the unconditional probability p2.
We can represent this observation formally by de ning,for any probability measure
q 2 ( ) and player i 2 N,an event [q]i = f!,pi(!jti(!)) = qg,Then,in this game,
[p2]2 =,By way of comparison,it is easy to see that it cannot be the case that [p1]1 =
(why?) regardless of how we specify p1.
Now take the point of view of Player 1,Since [p2]2 =,it is trivially true that
p1([p2]2jt1(!)) = 1 8!2 ;
in words,at any state !,Player 1 is certain that Player 2’s beliefs are given by p2 (i.e,by
),By the exact same argument,at any state !,Player 2 is certain that Player 1 is certain
that Player 2’s beliefs are given by p2,and so on and so forth.
I wish to draw your attention to two key conclusions that can be drawn based on this
analysis.
1,The standard model of games with payo uncertainty is capable of generating
in nite hierarchies of interactive beliefs about the underlying state of the world.
Indeed,this was precisely Harsanyi’s original objective,he realized that,in the presence
of payo uncertainty,the players’ strategic reasoning necessarily involves this sort of in nite
regress,and devised a very clever way to generate this information in a compact,manageable
way.
5
2,The standard de nition of Bayesian Nash equilibrium is deceivingly simple:
it hides a whole in nite hierarchy of implicit assumptions about higher-order
beliefs.
It is not my objective to argue whether or not these assumptions are reasonable,Rather,
I wish to emphasize that,just because they are not apparent in De nition 2 (or in any
textbook de nition of Bayes Nash equilibrium,for that matter),one should not conclude
that these assumptions are not necessary!
On the other hand,one should not assume that these assumptions are the only ones
consistent with Bayesian Nash equilibrium analysis,In fact,relaxing them leads to interesting
modelling possibilities|as I shall illustrate in the next lecture.
6
Lecture 4,Games with Payo Uncertainty (1)
Marciano Siniscalchi
September 28,1999
Introduction
The vast majority of games of interest in economics,nance,political economy etc,involve
some form of payo uncertainty,A simple but interesting example is provided by auctions:
an object is o ered for sale,and individuals are required to submit their bids in sealed
envelopes,The object is then allocated to the highest bidder at a price which depends on
every bid,according to some prespeci ed rule (e.g,a \ rst-price" or \second-price" rule),In
many circumstances (e.g,mineral rights auctions) it is reasonable to assume that the value
of the object is not known to the buyers,but that they receive some signal correlated with
it,In other cases,each buyer knows the value of the object to her,but not the other players’
valuation.
We can model this situation assuming that the set of actions available to every player
is the nonnegative real half-line,or some subset of it,representing allowable bids,While a
pro le of bids speci es the \outcome" of the auction,i.e,who receives the object and what
price she pays for it,it does not specify the payo to the winning bidder.
Her payo is determined by the (uncertain) value of the object; the uncertainty may even
be resolved only after the bidding game is over (e.g,when drilling or mining actually begins,
in the \mineral rights" case).
Thus,we need to extend our basic model of simultaneous games in order to account for
payo uncertainty.
The second issue is related to the solution concept(s) to apply in this setting,In the
auction example,it is reasonable to assume that players will condition their bid on whatever
signal they receive,In particular,in the \mineral rights" case,for any pro le of bids,players
cannot compute their actual payo because of the underlying uncertainty,but they can at
least compute their conditional expected payo given their signal.
This implies that a solution concept for games with payo uncertainty should specify an
action (or a set of actions) for each player and for each realization of whatever signal that
1
player may observe.
However,several subtle issues arise regarding the interpretation of \signals." If the
latter can be literally taken to be physical observations (e.g,samples or geological surveys),
then it makes sense to assume that players share a common probabilistic description of
the underlying uncertain quantities and their correlation with each player’s signal,Such
description could even be said to be part of the formulation of the game.
If,however,signals do not have a readily available physical interpretation (consider the
value,i.e,the maximum amount one is willing to pay for a painting,which presumably
includes a subjective element),then it does not make much sense to postulate that players
agree on any given probabilistic model of the underlying uncertain quantities|let alone make
such description part of the formulation of the game,Rather,a speci cation of players’ beliefs
about the underlying uncertainties should be considered as part of the solution concept.
Further observations will be in order once we specify a formal model of games with payo
uncertainty.
Games with Payo Uncertainty
I follow OR (with minimal expository deviations) and begin by providing a rather general
de nition.
De nition 1 A ( nite) normal-form game with payo uncertainty is a tupleG = (N; ;(Ai;ui;Ti)i2N),
where N is a nite set of players,is a set of states of the world,and for each i2N,Ai is
a set of actions,ui,Ai A i !R is Player i’s payo function,and Ti is a partition of
,referred to as Player i’s type partition.
It may be helpful to relate normal-form games with payo uncertainty with the decision-
theoretic framework introduced in Lecture 1,Taking the perspective of Player i,the set of
states of nature for i is ~ = A i ; the set of acts is (isomorphic to) Ai,with each act
ai 2Ai mapping a state (a i;!) 2 ~ to the outcome determined by the tuple (ai;a i;!);
nally,Player i’s preferences over outcomes are represented by ui.
Thus,Player i is not only uncertain about her opponents’ actions,she is also uncertain as
to the prevailing state of the world !2,In keeping with our general framework,a solution
concept should specify a probability distribution over whatever a player is uncertain about;
hence,it must also specify her beliefs about the prevailing state of the world.
Note,the usual \textbook" presentation of the situations referred to herein uses the
terminology \games with incomplete information,and includes a speci cation of a common
prior over the set of states of the world as part of the description of the strategic situation.
We use the somewhat nonstandard term \payo uncertainty" to emphasize that we do not
wish to regard probabilities over states of the world as part of the model,but rather as part
2
of the solution to the model,In particular,we do not wish to assume that such probabilities
are commonly agreed upon among the players,Notice that OR follows the same approach.
The key elements are the set of states of the world,which also a ect payo s (otherwise
there could be no payo uncertainty!),and the players’ type partitions.
The idea is that,if the prevailing state is !2,each player i2N is only noti ed that
the prevailing state is in the unique cell ti2Ti such that !2ti; that is,Player i rules out
any !062ti,but regards any !02ti as possible,We denote by ti(!) the set of states that
Player i deems possible at !2,that is,ti(!) is the unique ti2Ti such that !2Ti.
Indeed,in order to interpret the partition Ti correctly,one should think of states in each
cell ti2Ti as virtually indistinguishable from the point of view of Player i,This will be clear
from our use of the model.
To x ideas,consider the following version of the \mineral rights" auction setting,there
are two bidders,i = 1;2,and each receives a signal si2f0;1;:::;10g; the value of the object
(concession) is v(s1;s2) = s1 +s2.
Thus,=f0;1;:::;10g f0;1;:::;10g; Ti =ff(s1;s2),si = kg,k = 0;1;:::;10g; and
ui(ai;a i;(s1;s2)) = s1 +s2 ai if ai >a i,12(s1 +s2 ai) if ai = a i,and 0 otherwise.
The key observation is that each bidder i2N only observes her \part" of the true value,
si,On the other hand,both her signal and her opponent’s signal determine payo s.
Elements of the partition Ti are also referred to as types,Each type arguably captures a
possible information state of Player i,it tells us all she knows about the prevailing state of
the world.
Note well that,on the other hand,a type generally does not uniquely determine a player’s
payo corresponding to each action pro le,this is illustrated in the mineral rights auction
example above,Perhaps more interestingly,one could think of di erent information states,
sayti;t0i2Ti,such that,for all!;!02ti[t0i,ui(ai;a i;!) = ui(ai;a i;!0) for all (ai;a i)2A:
for example,this might be the case if,at ! and !0,player i attaches the same value to a
painting,but holds di erent beliefs as to the prevailing state of the world (hence,about her
opponents’ valuation),More on this point later.
Bayesian Nash Equilibrium
The standard analysis of normal-form games with payo uncertainty (suitably extended to
accommodate our decision-theoretic view) is centered upon the notion of Bayesian Nash
Equilibrium,due to John Harsanyi:
De nition 2 A Bayesian Nash Equilibrium of a normal-form game with payo uncertainty
G = (N; ;(Ai;ui;Ti)i2N) is a tuple (pi;( i;ti)ti2Ti)i2N such that,for all i2N:
(1) pi2 ( ) and,for all ti2Ti,pi(ti) > 0;
3
(2) For all ti2Ti and for all a0i2Ai,
X
!2ti
pi(!)
pi(ti)
8<
:
X
a i2A i
Y
j6=i
j;tj(!)(aj)
!" X
ai2Ai
i;ti(ai)ui(ai;a i;!)
!
ui(a0i;a i;!)
#9=; 0
Note that it is of course su cient to check for pro table deviations to pure actions.1
The product in parentheses is the probability that the action pro le a i will be played
in state !2,This is the product of the probability that each aj,j6= i,gets played,given
that,in state !,player j6= i is of type tj(!).
Next,the expression in curly braces is the di erence in expected payo from playing ai
and a0i in state !.
Finally,these di erences are weighted by the posterior probability that the state is actu-
ally !,given that the true state must lie in ti.
Observe that,it is also possible to formulate the best-reply property for type-contingent
randomized actions i = ( i;ti)i2N,As long as condition (1) above holds,it can be shown
that the resulting de nition is actually equivalent to the one given above,I will ask you to
provide the details.
Where does this come from?
We now take a step back and try to provide a rationale for this seemingly complex construc-
tion (due largely to John Harsanyi,a recent Nobel laureate),It turns out that,indeed,this
framework greatly simpli es the problem of describing strategic interaction in games with
payo uncertainty.
To keep the exposition simple (and in view of the fact that auctions will be the subject
of a forthcoming lecture),let us consider a simple two- rm Cournot model with uncertainty
about Firm 1’s cost.
Denote the latter variable by c2f0; 12g,Recall the setup,demand is given by P(Q) =
2 Q and each rm can produce qi2[0;1],Firm 2’s cost is zero.
It is probably easiest to think in standard equilibrium terms,Suppose that we are told
that a given tuple (q1;0;q1;1
2;q2) is \an equilibrium outcome" of the game,Let us ask ourselves
exactly what this implies.
At rst blush,there is little action on Firm 1’s side,she (?) expects Firm 2 to produce
q2 units,as per the equilibrium prescription,and best-responds to this,q1;c = BR1(q2;c) =
(1 c2) 12q2,for c2f0; 12g.
Now let us concentrate on Firm 2,Clearly,the latter cannot compute a best reply to
his opponent’s actions without formulating a conjecture about the relative likelihood of the
1Also note that,strictly speaking,what is de ned above is Bayesian Nash equilibrium for the mixed
extension of the original game G.
4
events c = 0 and c = 12,Denote by the probability of the former event; then,since Firm
2’s best reply is BR2( q1;0 + (1 )q1;1
2
),if we are given the information that q2 is a best
reply,we can \reverse-engineer" the value of (in this case,it is unique).
In terms of our model,we could let =f0; 12g,T1 =ff0g;f12gg,T2 =f g and p2(0) =
,It is easy to see that specifying p1 is not relevant for the purposes of De nition 2:
what matters there are the beliefs conditional on each t1 2T1,but these will obviously be
degenerate.
Let us think about the value of the parameter (or,equivalently,of the probability
p2),We have just noted that,given the equilibrium outcome,it is possible to derive Firm
2’s conjecture,Hence,since Firm 1 expects Firm 2 to play q2 and (according to the
equilibrium assumption) believes that Firm 2 expects Firm 1 to play q1;c,for each value
of the cost parameter c,it follows that Firm 1 can infer that Firm 2’s assessment of the
probability that c = 0 is,In other words,implicit in the speci c equilibrium we are looking
at is an assumption about Firm 1’s beliefs regarding Firm 2’s beliefs,It is easy to see that
we can continue in this fashion to build a whole hierarchy of interactive beliefs.
In terms of our model,note that Player 2’s type partition is degenerate,As a consequence,
at any state ! 2,Player 2’s conditional beliefs about,pi( jt2(!)) are the same|they
are given by the unconditional probability p2.
We can represent this observation formally by de ning,for any probability measure
q 2 ( ) and player i 2 N,an event [q]i = f!,pi(!jti(!)) = qg,Then,in this game,
[p2]2 =,By way of comparison,it is easy to see that it cannot be the case that [p1]1 =
(why?) regardless of how we specify p1.
Now take the point of view of Player 1,Since [p2]2 =,it is trivially true that
p1([p2]2jt1(!)) = 1 8!2 ;
in words,at any state !,Player 1 is certain that Player 2’s beliefs are given by p2 (i.e,by
),By the exact same argument,at any state !,Player 2 is certain that Player 1 is certain
that Player 2’s beliefs are given by p2,and so on and so forth.
I wish to draw your attention to two key conclusions that can be drawn based on this
analysis.
1,The standard model of games with payo uncertainty is capable of generating
in nite hierarchies of interactive beliefs about the underlying state of the world.
Indeed,this was precisely Harsanyi’s original objective,he realized that,in the presence
of payo uncertainty,the players’ strategic reasoning necessarily involves this sort of in nite
regress,and devised a very clever way to generate this information in a compact,manageable
way.
5
2,The standard de nition of Bayesian Nash equilibrium is deceivingly simple:
it hides a whole in nite hierarchy of implicit assumptions about higher-order
beliefs.
It is not my objective to argue whether or not these assumptions are reasonable,Rather,
I wish to emphasize that,just because they are not apparent in De nition 2 (or in any
textbook de nition of Bayes Nash equilibrium,for that matter),one should not conclude
that these assumptions are not necessary!
On the other hand,one should not assume that these assumptions are the only ones
consistent with Bayesian Nash equilibrium analysis,In fact,relaxing them leads to interesting
modelling possibilities|as I shall illustrate in the next lecture.
6
