Certain optimal strategies based upon game theory are given in this paper. A decision-making function and a risk function are explained. Decision-making criteria are applied for determining best decision-making functions with respect to a specific criterion. Special attention is given to the minimax criterion.saddle-point, minimax criterion, Bayesian criterion, loss function, risk function, decision-making function

Dominika Crnjac

Goran Martinovic

English

Research Papers in Economics

Interdisciplinary Management Research V 
SOME LINKS BETWEEN GAME THEORY AND DECISION 
THEORY IN ECONOMICS 
Dominika Crnjac
1, Goran Martinović
2
1Faculty of Electrical Engineering,  J.J. Strossmayer University of Osijek, Croatia, 
dominika.crnjac@etfos.hr
2Faculty of Electrical Engineering, J. J. Strossmayer University of Osijek, Croatia, 
goran.martinovic@etfos.hr
________________________________________________________________
Abstract
Certain optimal strategies based upon game theory are given in this paper. A decision-making 
function and a risk function are explained. Decision-making criteria are applied for determining 
best decision-making functions with respect to a specific criterion. Special attention is given to 
the minimax criterion. 
JEL classification: C02, C71, D81,
Key words: saddle-point, minimax criterion, Bayesian criterion, loss function, risk function, 
decision-making function 
________________________________________________________________
Introduction
In making a decision or a choice it seems rational to choose an option which 
has mathematical expectation "which promises most ", i.e. to choose an option 
which  minimizes  the  expected  loss  or  maximizes  the  expected  gain. 
Unfortunately,  this  simple  approach  to  decision-making  does  not  always 
function, since it is often very difficult to assign a numerical value either to a 
result or to probability of the outcome. This paper considers game theory and 
statistical games, and can be used as an introduction to decision theory. 
1. Game theory 
1.1 Zero-sum games 
The  game  is  said  to  be  a  zero-sum  game  when  there  is  a  conflict  between 
players, and whatever one player loses in the game the other player wins. It is 
possible to consider non-zero-sum games for more than two players, but they 
go beyond the scope of this paper. In game theory it is assumed that while 
165Interdisciplinary Management Research V 
selecting his/her strategy none of the players knows what the other player will 
do, noting that once the strategy is selected, it cannot be changed. 
Games are classified according to the number of strategies available to every 
player. For example, if every player has to choose between two strategies, then 
it is going to be a 2´ 2 game. If one player has 4 options, and the other 5, then 
it will be a 4 5 ´  or a 5 4 ´  game. 
In  this  paper  we  consider  just  those  games  in  which  every  player  has  only 
finitely many options.
1.1.1 Notation
Let us denote two players by Player A and Player B. 
Player A options will be denoted by I, II, III, etc. 
Player B options will be denoted by 1, 2, 3, etc. 
The game might be presented by a matrix, e.g. a 2 2 ´  game might be presented 
as follows: 
Matrix elements are called payoffs (i.e. loss for Player A and gain for Player B). 
Amounts L(I,1), L(II,1), L(I,2) and L(II,2) are loss function values of a certain 
game. L(I,1) represents loss of Player A (and gain of Player B) when Player A 
chooses option I, and Player B chooses option 1, etc. 
Game theory tries to find optimal strategies, i.e. the ones most profitable for 
every player, as well as a corresponding payoff or value of the game. 
1.1.2 Saddle-point
A  pair  of  strategies  will  be  in  balance  if  and  only  if  element  L(a,b)
corresponding to balance is at the same time the greatest in the column and the 
least in the row. Such strategy is called a saddle-point. 
For example, a 3 3 ´  matrix 
has a saddle-point in the second row and the second column, but a 2 2 ´  matrix 
L(I,1) L(II,1)
L(I,2) L(II,2)
6  2  2 
4  3  5 
-3  1  0 
I II
1
Player A
Player B
I II III
Player B 1
2
3
Player A
2
166Interdisciplinary Management Research V 
2 2
2 2
−  
  −  
does not have saddle-points. 
1.1.3 Domination
In a payoff matrix one strategy (represented by a row or a column) dominates
14
over the other strategy if the choice of the first strategy is at least as good as the 
choice of the second strategy and in some cases even better. Player A will look 
for a lower loss, whereas Player B will look for a greater gain. The dominated 
strategy can always be rejected in the game.
For example, in a ( ) 4 3 ´  game
Since Player A looks for a lower loss, strategy II dominates over strategy IV, 
since the loss of Player A will always be less if strategy II is selected and not 
strategy IV. Therefore, Player A will never choose strategy IV; hence, it can be 
rejected.
The game is reduced to a 3 3 ´  game 
However, since Player B looks for a greater gain, strategy 3 now dominates 
over strategy 1, since the gain of Player B is always greater if strategy 3 is 
selected and not strategy 1. Thus, the game is reduced to a 3 2 ´  game
14 R.B.Myerson; (1997), p.26, Game Theory Analysis of Conflict, First Harvard University 
Press (paperback edition) 
2  0  1  4 
1  2  1  4 
4  1  3  2 
2  0  1 
1  2  1 
4  1  3 
1  2  1 
4  1  3 
I II III IV
Player B 1
2
3
Player A
I II III
1
2
Player A
I II III
Player B 2
Player A
3
Player B
3
167Interdisciplinary Management Research V 
2. Statistical games 
By observing social, physical and other phenomena, regulating laws or relations 
are called state of nature. The conclusion about the state of nature is reached on 
the basis of results obtained from a statistical experiment. Drawing a conclusion 
about the state of nature on the basis of experimental results is called statistical 
inference. A statistical inference problem can be considered a two-player game.
In  statistical  inference  decisions  on  the  population,  such  as  expectation  or 
variance of some property, are based upon sample data. Statistical inference 
might  therefore  be  observed  as  a  game  between  the  Nature  that  controls 
relevant  features  of  the  population  and  a  statistician  who  tries  to  make  a 
decision on the population. 
One way in which statistical games differ from game theory is that in game 
theory every player selects a strategy without knowing what the opponent will 
do. In a statistical game, a statistician has sample data which provide some 
information about the choice of nature. 
The state of nature denoted by θ  is unknown to a statistician. The set of all 
states of nature is denoted by Ω. A statistician will take some action a or make 
a decision a if he/she finds out that the state of nature is  θ . The set of all 
actions or decisions is denoted by A. Prior to taking some action, a statistician 
conducts an experiment for the purpose of collecting data on the state of nature. 
Experiment outcome is a random variable whose probability law depends on an 
unknown parameter. If in one experiment a random variable C  takes the value 
x, a statistician takes an action, i.e. makes a decision  ( ) a d x = . Function  ( ) d x
is  called  a  decision-making  function.  By  this  function  decision  ( ) a d x =   is 
unambiguously assigned to every experiment outcome x.
Example 1. 
A statistician is told that a coin is either normal or double-headed. A statistician 
cannot examine the coin but he/she can notice what comes up after tossing the 
coin. Then, a statistician should decide whether the coin is double-headed or 
not. If a statistician makes a wrong decision, he/she should pay a fee in the 
amount of 1 CU
15. In case of a right decision there is no fee (or reward). 
Ignoring the fact that a statistician observes one tossing, the problem can be 
considered as the following game 
15 CU - currency unit 
168Interdisciplinary Management Research V 
1 2 a a
where
1 θ = "state of nature" is that the coin is double-headed 
2 θ =  "state of nature" is that the coin is balanced
1 a =  decision made by a statistician is that the coin is double-
headed
2 a = decision made by a statistician is that the coin is balanced
However,  a  statistician  knows  what  happened  at  tossing,  i.e.  a  statistician 
knows whether a random variable  X  took the value  0 x=  (heads) or  1 x=
(tails). A statistician wants to use that information for making a choice between 
1 a  and  2 a , so that the decision-making function determining which action to 
take in any of the cases is 
( )
1
1
2
  0
  1
a when x
d x
a when x



,    =
=
,    =
,
i.e. ( ) ( ) 1 1 1 2 0   1 d a and d a =      = .
The  purpose  of  indices  is  to  distinguish  different  functions.  Other  possible 
decision-making functions might be
( ) 2 1 0 d a =  and  ( ) 2 1 1 d a =
i.e., always choose  1 a  regardless of the experiment outcome, 
( ) ( )
( ) ( )
3 2 3 2
4 2 4 1
0 1
  0 1
or d a and d a
or d a and d a
   =       =
  =       =
1
2
θ
θ
Nature 0 1
(Player B) 1 0
Statistician
(Player A)
169Interdisciplinary Management Research V 
The following table gives corresponding values of the loss function 
1 2 a a
1
2
θ
θ
( )
( )
1 1
1 2
,
,
L a
L a
θ
θ
( )
( )
2 1
2 2
,
,
L a
L a
θ
θ
One option is to select  1 a  when 0 x=  and  2 a when 1 x= , which might be 
expressed as
( ) ( ) ( ) 1 1 , , j j R d E L d x θ θ   =  ,
whereby ( ) ( ) 1 , j L d x θ      is a loss function in case of the decision  ( ) 1 d x  and 
Nature strategy  j θ .
Expectation is calculated with respect to a random variable x and 
        with ( ) 1   0 1 P x θ         = =
                        ( ) 1 0 P x= =
             with  ( ) 2 0 1/2 P x θ            = =
                           ( ) 1 1/2 P x= =  
The aforementioned gives 
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
1 1 1 1 2 1
1 2 1 2 2 2
2 1 1 1 1 1
2 2 1 2
, 1 , 0 ,
, 1/2 , 1/2 ,
, 1 , 0 ,
, 1/2 ,
R d L a L a
R d L a L a
R d L a L a
R d L a
θ θ θ
θ θ θ
θ θ θ
θ θ
+
+
+
+
 =                   = 1´ 0+ 0´ 1            = 0
 =          = 1/ 2´ 0+ 1/ 2´ 0    = 1/ 2
 =                   = 1´ 0+ 0´ 0            = 0
 =    ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1 2
3 1 2 1 2 1
3 2 2 2 2 2
4 1 2 1 1 1
1/2 ,
, 1 , 0 ,
, 1/2 , 1/2 ,
, 1 , 0 ,
L a
R d L a L a
R d L a L a
R d L a L a
θ
θ θ θ
θ θ θ
θ θ θ
+
+
+
      = 1/ 2´ 1+ 1/ 2´ 1     = 1
 =                   = 1´ 1+ 0´ 1             = 1
 =          = 1/ 2´ 0+ 1/ 2´ 0    = 0
 =                   = 1´ 1+ 0´ 0   
( ) ( ) ( ) 4 2 2 2 1 2 , 1/2 , 1/2 , R d L a L a θ θ θ +
          = 1
 =          = 1/ 2´ 0+ 1/ 2´ 1     = 1/ 2
Let us note that a 4 2 ´  game is a zero-sum game for two players where payoffs 
are equal to corresponding values of the risk function. 
Statistician
Nature
170Interdisciplinary Management Research V 
                 Statistician 
1 2 3 4 d d d d                     
    Nature 
1
2
θ
θ
Note that  1 d  dominates over  2 d , and  3 d  dominates over  4 d , so that  2 d  and  4 d
can be rejected. 
The game is reduced to a 2 2 ´  zero-sum game for two players. 
              Statistician 
1 3 d d
         Nature         
1
2
θ
θ
An optimal strategy for a statistician is with probabilities  2/3 and 1/3 for  1 d
and 3 d , respectively. 
3. Decision-making criteria 
Generally speaking, it is possible to find the best decision-making function only 
with respect to some criteria. In the sequel we will consider two criteria. 
Minimax criterion 
According to the minimax criterion, a decision-making function is chosen for 
which ( ) , R d θ , maximized by θ , is minimal. 
If we apply the minimax criterion to the example from Section 2 with  2 d  and 
4 d , maximal risk for  1 d  is 1/2, and for  3 d  it is 1. Thus,  1 d  minimizes maximal 
risk.
Bayesian criterion 
If Q  is  observed  as  a  random  variable,  in  accordance  with  the  Bayesian 
criterion,  a  decision-making  function  is  chosen  for  which  ( ) , E R d θ       is 
minimal, where expectation is calculated with respect to Q. For the criterion it 
is necessary to consider Q as a random variable with the given distribution.
0  0  1  1 
1/2 1  0  1/2
0  1 
1/2 0
171Interdisciplinary Management Research V 
Application of the Bayesian criterion to the example from Section 2 requires 
probabilities to be assigned to strategies of the Nature  1 θ  and  2 θ . If  ( ) 1 P p θ =
and ( ) 2 P p θ = 1- , then the Bayes risk is for
1 d = ( ) ( ) 0 1/2 1 1/2 1 p p p ⋅ + − − =
3 d = ( ) 1 0 1 p p p ⋅ + − = .
When 1/3 p> , Bayes risk as to  1 d  is less than Bayes risk for  3 d ,  so that  1 d  is 
preferred over  3 d .
When 1/3 p< , Bayes risk as to  3 d  is less than Bayes risk for  1 d , so that  3 d  is 
preferred over  1 d .
When 1/3 p = , both Bayesian criteria are equal and any  1 d  and  3 d  can be 
chosen.
The aforementioned can be improved such that the problem is translated to the 
basic set ↔ sample, so that decision-making is of the form: 
a) A set of all possible values is defined that can be taken by Q in the problem 
under consideration. This set is denoted by Ω and it is called parameter space.
b) A set of all possible actions or decisions is defined, which can be reached in 
the problem under consideration. This set is called action space or decision 
space and it is denoted by A .
c) We define function
( ) 1 2 , ,..., n a d x x x =
of a random sample  1 2 , ,... n x x x , which is called a decision-making function.. A 
set of values of this function is action space A. This means that a statistician 
takes an action  a, if for the realized value of the sample  1 2 , ,... n x x x  he/she has 
obtained ( ) 1 2 , ,..., n a d x x x = .
Example 2.  Let  ( ) ; f x Q  be the Gaussian probability distribution law
16
( )
( )
2
2 ; , ,
x
f x e x
π
−Θ
−
−∞ < < +∞
1
Q =
2
where Q is any real number. 
Parameter space Q is a set of real numbers 
{ }. Ω ∞ ∞ = Q½ - < Q< +
16 Sarapa, N.; (1988), p.265, Teorija vjerojatnosti., Zagreb, Školska knjiga, UDK 519.21(075.8) 
172Interdisciplinary Management Research V 
If for different values of  Q we take different actions, the set of actions  A  is 
also a set of real numbers
{ } a a ∞ ∞ A = ½ - < < + .
For a decision-making function we can take any function of the random sample 
1 2 , ,... n x x x  taken from this normal arrangement. For example, we can take the 
arithmetic mean
17 of the sample
( ) 1 2
1
1
, ,... .
n
n i
i
d x x x x
n = ∑ =
For the realization of sample  1 2 , ,... n x x x  we obtain 
( ) 1 2
1
1
a , ,... .
n
n i
i
d x x x x
n = ∑ = =
Loss function
In the game against the nature, a statistician will have gain or loss, depending 
on the action taken i.e. decision made. Since the action
  Table 1:  
          possible actions 
1 a 2 a
states
1 Θ
of  nature  
2 Θ
0 10
     10                           0
In order to find something out about the state of nature, a statistician conducts 
an experiment whose outcomes are values of a random variablex. Let a random 
variable X  take the value  0 x=  in case of heads, and the value  x= 1 in case 
of  tails.  In  order  to  use  the  information  obtained  through  the  experiment,  a 
statistician must define all possible decision-making functions.
17 Crnjac D., Crnjac M.; (2004), Nejednakosti, razlike i odnosi meñu nekim statističkim
sredinama, Zbirka radova XIII., Dane Kordić, Frano Ljubić, Ivan Pavlović, Brano Markić, Ante 
Markotić, Dražena Tomić, Sanja Bijakšić(ur), pp.171-186, UDK 33 (06.055.2), Ekonomski 
fakultet u Mostaru, Sveučilište u Mostaru 
173Interdisciplinary Management Research V 
One possibility is to take action  1 a  when  0 x=  and action  2 a , when  x= 1, that 
defines the decision-making function
( )
1
1
2
,    0,
,    1,
a when x
d x
a when x



=
=
=
whose values are  ( ) ( ) 1 1 1 2 a     a . d and d 0 = 1 =
The other possibility is to take action  1 a  in both cases, so that the decision-
making function is 
( )
1
2
1
,    0,
,    1,
a when x
d x
a when x



=
=
=
with values  ( ) ( ) 2 1 2 1     . d a when d a 0 = 1 =  
It is possible to define two more decision-making functions 
( )
2
3
2
,    0,
,    1,
a when x
d x
a when x



=
=
=
whose values are  ( ) ( ) 3 2 3 2     ,  d a and d a and 0 = 1 =  
( )
2
4
1
,    0,
,    1,
a when x
d x
a when x



=
=
=
with values  ( ) ( ) 4 2 4 1     . d a and d a 0 = 1 =
As it can be seen, the decision-making function assigns one action to every 
value of a random variablex. Since we have two values of a random variable 
x, i.e. 0   1 and , and two possible actions  1 a  and  2 a , there are 
2 2 4 =  decision-
making  functions  that  are  previously  denoted  by 
( ) ( ) ( ) ( ) 1 2 3 4 ,  ,   . d x d x d x and d x      Values of these functions can be represented 
by Table 2. 
Table 2: 
d1 d2 d3 d4
a1 a1 a2 a2
a2 a1 a2 a1
0
1
x
x
Choice of the loss function  ( ) , L a Q  depends on the nature of the problem under 
consideration. In problems referring to statistical evaluation, a loss function is 
taken to be of the form 
( ) ( )
2 , L a a− Q = Q ,
174Interdisciplinary Management Research V 
and  it  is  called  a  quadratic  loss  error
18.  In  problems  referring  to  testing 
statistical hypotheses of loss functions, function
( )
0, if a right decision has been made,
,
1, if a wrong decision has been made
L a



Q =
Conclusion
Some optimal strategies, i.e. decision-making functions, are given in this paper. 
Decision-making  criteria  as  well  as  criteria  for  optimizing  decision-making 
functions are also given with respect to some criteria, and certain improvements 
are made. 
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Ljubić, Ivan Pavlović, Brano Markić, Ante Markotić, Dražena Tomić,
Sanja Bijakšić(ur), pp.171-186, UDK 33 (06.055.2), Ekonomski fakultet 
u Mostaru, Sveučilište u Mostaru 
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Some Links Between Game Theory and Decision Theory in Economics

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Some Links Between Game Theory and Decision Theory in Economics

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