TADEUSZ OSTROWSKI
ON MIXED ASSURANCE CHOICE IN DECISIONS UNDER UNCERTAINTY Summary
In the paper we focus on a mixed assurance choice or in other words zero risk mixed choice in some decisions problems under uncertainty. There are given neces-sary and sufficient conditions for the existence of this kind of choice which in the language of game theory states for the mixed Nash equilibrium in a symmetric game. The conditions and formulas for computing the zero risk mixed choice are ex-pressed with saddle point matrices.
Keywords: Decision Under Uncertainty, Criteria of Choice, Nash Equilibrium, Zero Risk Mixed Choice
1. Introduction
The aim of the paper is showing an algebraic approach to some cases of decision making un-der uncertainty. This approach is based on the concept of saddle point matrices.
One of the most characteristics to be successful is the ability to make the correct decisions. Game theory, as an important part of economics, provides a framework to study interactions between intelligent agents or when an agent plays against nature (considered as the agent not interested in winning). In the first case the agents are modeled as players, called Row and Column for convenience, with partially or completely conflicting interests, who make simultaneous deci-sions among different strategies (action spaces), aiming to maximize regards (payoffs).
What is the best decision for the agent, engaged in a game against nature, depends on what in-formation the agent has regarding how nature “chooses” its actions from a nature state space. It is assumed that the agent does not know the precise nature action to be chosen. It is also assumed that there are given a set of acts from which the decision maker has to choose, a set of potential states and that one and only one of the states will be realized.
From the agent's perspective, there are two alternative possible models that can be used for na-ture: the first one is nondeterministic and represents the heart of decision theory, where the agent has no idea what nature will do, so in the literature it is often called the horse lottery because of reference to the lack of knowledge which horse (strategy) to choose and which horse (state of nature) will be the winner. The second model is probabilistic, where the agent has been observing nature and gathering statistics.
In the paper the nondeterministic model will be focused. When the act has been chosen and the state has been realized, the payoff is determined by the Rn, n matrix
A =
nn n n
a
a
a
a
...
...
...
...
...
1 1 11 .Rows of the matrix A constitute outcomes for possible acts, denoted by ri (i = 1, …, n), and columns constitute potential states, denoted by si (i = 1, …, n). According to that aij means a payoff attainable under the j-th state of nature (Column player) when the decision maker (Row player) has taken the i-th pure strategy.
2. Choice Criteria
There are many different criteria of choosing the best strategy if you have to make a decision under uncertainty. The best known are Wald criterion, maximax one, Hurwicz one, regret one or Laplace one. They have some advantages like being independent form linear change of scale, identifying the optimal decision in most cases, or giving the possibility to order decisions from the best to the worst. But they have disadvantages as well; in some cases they lead to the paradox of choice identifying different strategies or they give a set of optimal decisions instead only unique one. In a brief review we remind these criteria.
Wald Criterion (Maximin Criterion or Maximin Gain Rule). In this pessimistic approach, the decision-maker selects that strategy which is associated with the best possible worst outcome, In other words, the criterion suggests the decision maker examine only the minimum payoffs of alternatives and choose the alternative whose outcome is the least bad. This assurance and con-servative criterion – according to the philosophy a bird in the hand is worth two in the bush – appeals to the cautious decision maker who seeks to ensure that in the event of an unfavorable outcome, there is at least a known minimum payoff. Formally
strategyoptimal = r
max
( rjj
a
min
); r, j = 1, …, n. (1)The maximax criterion. The maximax criterion – opposite to Wald criterion – represents an optimistic approach. It suggests that the decision maker should examine the maximum payoffs of alternatives and choose the alternative whose outcome is the best. This criterion appeals to the adventurous decision maker who is attracted by high payoffs or likes to gamble and who is in the position to withstand any losses without substantial inconvenience. Formally
strategyoptimal = r
max
( rjj
a
max
); r, j = 1, …, n. (2)Hurwicz Criterion (The Middle of The Road Rule). This approach attempts to strike a bal-ance between the maximax and maximin criteria. It suggests that the minimum and maximum of each strategy should be averaged using Ȝ and 1– Ȝ as weights. The parameter Ȝ, called the coeffi-cient of realism, represents the index of optimism and the alternative with the highest average is selected. In other words, Ȝ reflects the decision maker’s attitude towards risk taking and states for subjective evaluation of the probability of the best outcome. Notice that a cautious decision maker will set Ȝ = 0 which reduces the Hurwicz criterion to the maximin criterion. An adventurous
decision maker will set Ȝ = 1 which reduces the Hurwicz criterion to the maximax criterion. Formally sr = Ȝ rj j
a
max
rj ja
min
+ (1– Ȝ) rj ja
min
(r, j = 1, …, n), strategyoptimal = r rs
max
(r = 1, …, n).Savage Criterion (Minimax Regret Criterion). The model is based on a regret matrix which compares, more precisely subtracts, the highest outcomes of each strategy from other outcomes. The new matrix shows the extent to which a decision maker could have done better (opportunity loss). Savage's regret-based decision-model avoids the extreme conservatism of the maximin criterion because Wald rule is applied to this new matrix to gain the minimax regret solution. This means that Savage criterion can be interpreted as Wald criterion with respect to the regret matrix. Formally, the regret corresponding to a particular payoff aij is defined as
rij = ( ij i
a
max
) – aij, and strategyoptimal = imin
( rj jr
max
); i, j = 1, …, n. (4)It can be noticed that definition of regret allows the decision maker to transform the payoff matrix into a regret matrix, and the criterion suggests that the decision maker look at the maximum regret of each strategy and select the one with the smallest value. This approach appeals to cau-tious decision makers who want to ensure that the selected alternative does well when compared to other alternatives regardless of what situation arises.
Laplace Criterion (Equal Likelihood Criterion, The Insufficient Reason Criterion). This approach postulates that if no information is available about the probabilities of the various out-comes, it is reasonable to assume that they are all equally likely. Therefore, if there are n outcomes, the probability of each is equal to
n
1
and the decision maker should calculate the expected payoff for each alternative and select the alternative with the largest value. The use of expected values distinguishes this approach from the criteria that use only extreme payoffs and makes the approach to be very similar to decision making under risk. Formally
sr =
n
1
= n k rk
a
1 , strategyoptimal = r rs
max
(r = 1, …, n). (3) (5)3. Zero Risk Mixed Choice
(MC for short). Denote by e∈Rn the column vector with all components of ones, and let E∈Rn, n
stands for the matrix with all entries equal to one. Hence E = eeT. For a given square matrix A∈Rn, n denote by (A, e) the matrix, often called the saddle point matrix, and defined as follows
(A, e) =
0
Te
e
A
. (6)The matrices obtained from A by replacing the k-th row or column, where k = 1, …, n, with the row vector eT or the column vector e are denoted by Ak or Ak, respectively. For a review of some properties of saddle point matrices with two vector blocks1. Saddle point matrices with all matrix blocks are considered in Benzi, Golub and Liesen2.
A bimatrix, or finite two-player strategic game is given by an ordered pair of payoff matrices, A = [aij] and B = [bij], with equal dimensions. When the row and column players choose their i-th and j-th pure strategies, respectively, the row player’s payoff is aij and the column player’s payoff is bij. A mixed strategy is a probability vector x specifying the probability with which each pure strategy is played. If these probabilities are all strictly positive, x is said to be completely mixed. A completely mixed Nash equilibrium is one in which both players’ strategies are completely mixed. This paper concerns games in which the number of pure strategies is the same for both players, so that A and B are square, n × n matrices (n 2). Furthermore, in next part of the paper it is assumed that B = AT what implies we deal with bimatrix symmetric games and in the decision problem under uncertainty the column player states for nature.
Recall that for bimatrix games [A, B] a pair of strategies (p, q) is said to be Nash equilibrium if
pTAq xTAq for all strategies x of the first player, pTBq pTBy for all strategies y of the second player.
In a particular case, i.e. in symmetric bimatrix games [A, AT], a profile p is a Nash equilibrium (NE) if and only if
pTAp xTAp for all strategies x.
Theorem 1. For any A, B∈Rn, n, where det(A, e), det(B, e) 0, the bimatrix game [A, B] – in which players strategies are completely mixed – has the players equilibrium payoffs E(A), E(B) equal to E(A) = –
)
,
det(
det
e
A
A
, E(B) = –)
,
det(
det
e
B
B
(8) and components of the equilibrium profile vectors p, q are as follow:1
see Ostrowski T. (2007). On Some Properties of Saddle Point Matrices with Vector Blocks, International Journal of Algebra, Vol. 1, no. 3, p. 129–138.
2
Benzi M.. Golub G. H. and Liesen J. (2005). Numerical solution of saddle point problems, Acta Numerica, Cambridge University Press, p. 1–137.
pi =
)
,
det(
)
,
det(
e
e
B
B
i , qi =)
,
det(
)
,
det(
e
e
A
A
i ; i = 1, …, n. (9)The proof of the theorem above can be found in Milchtaich and Ostrowski3. Another proof, based on the concept of algebraic cofactors and their special sums, has been provided in Os-trowski4. As a special case (B = AT) of the theorem we obtain immediately that in any symmetric bimatrix game [A, AT], A∈Rn, n, if players strategies are completely mixed, then the both players payoff E(A) equals
E(A) = –
)
,
det(
det
e
A
A
, (10)and components of the equilibrium profile vector p are pi =
)
,
det(
)
,
det(
e
e
A
A
i ; i = 1, …, n. (11)Theorem 2. The necessary and sufficient condition for existence of a unique completely mixed equilibrium in a symmetric bimatrix game [A, B] is that for all i, j = 1, …, n the following holds
det(Ai, e)det(Aj, e) > 0, det(Bi, e)det(Bj, e) > 0 .
The proof of the Theorem 2 can be found in Milchtaich and Ostrowski5. 4. Numerical Example
In the game against nature let be given a payoff matrix of the following form
A =
−
−
−
4
3
2
2
4
4
4
2
6
. (13)If a decision maker wants to apply Wald criterion, then by (1) we obtain strategyoptimal = r3 = (2 3 4).
If a decision maker decides to apply Maximax criterion, then by (2) we have strategyoptimal = r1 = (6 –2 –4).
3
Milchtaich I. and Ostrowski T. (2008). On some saddle point matrices and applications to completely mixed equilibrium in bimatrix games. International Journal of Mathematics, Algebra and Game Theory, Vol.18 (2008), pp. 1–8.
4
Ostrowski T. (2006). Population Equilibrium with Support in Evolutionary Matrix Games, Linear Algebra and its Applications, 417, p. 211–219.
5
Milchtaich I. and Ostrowski T. (2008). On some saddle point matrices and applications to completely mixed equilibrium in bimatrix games. International Journal of Mathematics, Algebra and Game Theory, Vol.18 (2008), p. 1–8.
If a decision maker wants to apply Hurwicz criterion for Ȝ = 0.75, then by (3) we get strategyoptimal = r1 = (6 –2 –4).
If a decision maker decides to apply Laplace criterion, then by (5) we obtain strategyoptimal = r3 = (2 3 4).
If a decision maker wants to apply Savage criterion, then by (4) the regret matrix R has the following form R =
0
1
4
6
0
2
8
6
0
, and strategyoptimal = r3 = (2 3 4).If a decision maker wants to apply MC, then the calculation goes as follows
det(A1, e) = –det
−
4
3
2
2
4
4
1
1
1
= –6, det(A2, e) = –det−
−
4
3
2
1
1
1
4
2
6
= –6, det(A3, e) = –det−
−
−
1
1
1
2
4
4
4
2
6
= –48, det(A, e) == 3 1
)
,
det(
i iA e
= 60, detA = det−
−
−
4
3
2
2
4
4
4
2
6
= 156.Therefore, by theorem 1, applying (10) and (11), we obtain
p =
10
1
8
1
1
, and E(A) = –)
,
det(
det
e
A
A
= 2.6.Table 1. Comparison of optimal choices
Criterion Choice E min ma
x MC r1 : r2 : r3 = 1 : 1 : 8 2.6 2.6 2.6 Maximax r1 6 – 4 6 Maximin r3 2 2 4 Hurwicz (Ȝ = 0.75) r1 3 – 4 6 Laplace r3 3 2 4 Savage r3 - 2 4
Source: Own work.
E 6 4 3 2.6 2 –4 1 Maximax Hurwicz Wald Laplace Savage
Mixed zero risk choice
Ȝ
Figure 1. Graphic presentation comparison of optimal choices Source: Own work.
Another example of application of game theory to decision problem under uncertainty (with using two criteria of choice, Wald criterion and Laplace criterion) can be found in Gharbani6.
6
Gharbani M. (2008). Application of Game Theory to Compare the Effect of Market Sale and Contract Strategies. World Applied Sciences Journal 4 (4), p. 596–599.
5. Conclusions
Zero risk mixed choice, called also mixed prudential choice, can be successfully applied when the decision maker is able to diversify available strategies. In the paper terms MC and mixed NE are used equivalently and a game against nature with the payoff matrix A is regarded as a symmet-ric bimatrix game [A, AT]. It is also well known that any finite symmetric game has a symmetric equilibrium.
The assumption that the matrix A is nonsingular may appear to be quite restrictive, since it is known that singular A appears in many applications (see for instance Haber and Ascher7).
However if A∈Rn, n is singular and rank A = n–1, then one can replace A by a nonsingular ma-trix ĮA + ȕE for some Į, ȕ∈R, and the matrix ĮA + ȕE does not change the optimizer (see also Bomze and de Klerk8) of the model with the payoff matrix A. Furthermore, letting Į = 1, ȕ = –Ȝ one can get the matrix associated with so called dual formulation of copositive program (for details see Bomze9).
Otherwise, one can use augmented Lagrangean techniques to replace (6) with an equivalent system in which the left-up block is nonsingular and it has the same solution (see Benzi, Golub and Liesen10).
It follows from the theorem 1 that a bimatrix game [A, B] such that (A, e) and (B, e) are nonsingular has at most one completely mixed equilibrium.
If (A, e) and (B, e) are singular, multiple equilibria may exist, possibly with different payoffs. However, multiplicity of equilibria is possible only if A or B, respectively, are also singular. For example, in the symmetric bimatrix game [A, AT] with the payoff matrix
A =
0
2
0
2
0
0
2
0
0
0
0
2
a
a
a
a
a
a
a
a
(a ∈ R+),it is easy to see that for any p∈(12, 1) the strategy [1 – p, p –21 ,1 – p, p –21 ]T gives complete-ly mixed equilibrium with the payoff equal to E(A) = ap.
If (A, e) is singular but A is nonsingular, then (12) cannot possibly hold, so that a completely mixed equilibrium does not exist. For example in the symmetric bimatrix game [A, AT] with the payoff matrix A =
a
a
a
a
3
2
2
(a 0)7 Haber E. and Ascher U. M. (2001). Preconditioned all-at-once methods for large, sparse parameter estimation problems,
Inverse Problems 17 (2001), p. 1847–1864.
8 Bomze I. M. and de Klerk E. (2002). Solving standard quadratic optimization problems via linear, semidefinite and
copositive programming, Journal of Global Optimization 24, pp. 163–185.
9 Bomze I. M. (1998). On standard quadratic optimization problems. Journal of Global Optimization 13, pp. 369–387. 10 Benzi M.. Golub G. H. and Liesen J. (2005). Numerical solution of saddle point problems, Acta Numerica, Cambridge
we have det
a
a
a
a
3
2
2
0, det(A, e) = det0
1
1
1
3
2
1
2
a
a
a
a
= 0.The reason of that a completely mixed equilibrium does not exist for the given payoff matrix A is that Row’s first pure strategy is dominated for a > 0, and the second pure strategy is dominat-ed for a < 0.
Therefore not every bimatrix game has a mixed NE. However, such equilibria always exist for certain classes of matrices (see Milchtaich and Ostrowski11 or Ostrowski12 for more details; some classes of bimatrix games with mixed NE are listed there).
If A is not a square matrix, and A∈Rm, n, then the method of Gale, Kuhn and Tucker for sym-metrisation can be used which leads to a square matrix. For instance if A∈Rm, n, then we obtain the square payoff matrix of the following form
−
−
−
0
T T T n m n mO
A
A
O
e
e
e
e
.The concept of NE has particular meaning in evolutionary matrix games because of applica-tions to biological and social sciences and because it is the necessary condition of stability the solution vector. From that point of view, MC can be interpreted as a potential stable point of a dynamic adjustment process in which individuals adjust their behavior to that of the other players in the game, searching for strategy choices that will give them better results (Holt and Rorh13).
The concept of MC has also some unpractical properties. One of them is Pareto non-effectiveness which occurs in some cases (for example, Prisoner’s dilemma, see Ostrowski14 for details). Therefore, the MC not always can be accepted as a solution of the game.
The paper complements and extends the author some earlier results. The computational prob-lem of finding NE has received much attention. Completely mixed equilibria in bimatrix games have been extensively studied. It is known that many characterizations of M-matrices can be deduced from the theory of completely mixed games. Since not every bimatrix game has a com-pletely mixed equilibrium, but such equilibria exist for certain classes of bimatrix games, it would be important to distinct all classes of bimatrix games with a mixed equilibrium.
11
Milchtaich I. and Ostrowski T. (2008). On some saddle point matrices and applications to completely mixed equilibrium in bimatrix games. International Journal of Mathematics, Algebra and Game Theory, Vol.18 (2008), p. 1–8.
12
Ostrowski T. (2007). A Necessary ad Sufficient Condition of a Mixed NE in Bimatrix Games, International Journal of Algebra Vol. 1, no. 7, p. 303–310.
13
Holt, Charles A. and Roth, Alvin. E. (2004). The Nash Equilibrium: a perspective. PNAS, Vol 101, no. 12, p. 3999–4002.
14
Ostrowski T. (2009). Cooperation of human as a strategy leading to success. The art of human resource management (Chapter 2). Ed. G. Drozdowski, Z. Dobrowolski, The State Vocational University in Gorzów Wlkp, Institute of Man-agement, p. 27–36.
%LEOLRJUDSK\
[1] Benzi M.. Golub G. H. and Liesen J. (2005). Numerical solution of saddle point problems, Acta Numerica, Cambridge University Press, p.1–137.
[2] Bomze I. M. (1998). On standard quadratic optimization problems. Journal of Global Optimi-zation 13, p. 369–387.
[3] Bomze I. M. and de Klerk E. (2002). Solving standard quadratic optimization problems via linear, semidefinite and copositive programming, Journal of Global Optimization 24, p. 163– 185.
[4] Gharbani M. (2008). Application of Game Theory to Compare the Effect of Market Sale and Contract Strategies. World Applied Sciences Journal 4 (4), p. 596–599
[5] Haber E. and Ascher U. M. (2001). Preconditioned all-at-once methods for large, sparse parameter estimation problems, Inverse Problems 17 (2001), p. 1847–1864
[6] Holt, Charles A. and Roth, Alvin. E. (2004). The Nash Equilibrium: a perspective. PNAS, Vol 101, no. 12, p. 3999–4002.
[7] Milchtaich I. and Ostrowski T. (2008). On some saddle point matrices and applications to completely mixed equilibrium in bimatrix games. International Journal of Mathematics, Al-gebra and Game Theory, Vol.18 (2008), p.1–8.
[8] Ostrowski T. (2006). Population Equilibrium with Support in Evolutionary Matrix Games, Linear Algebra and its Applications, 417, p.211–219.
[9] Ostrowski T. (2007). On Some Properties of Saddle Point Matrices with Vector Blocks, Inter-national Journal of Algebra, Vol. 1, no. 3, p. 129–138.
[10] Ostrowski T. (2007). A Necessary ad Sufficient Condition of a Mixed NE in Bimatrix Games, International Journal of Algebra Vol. 1, no. 7, p. 303–310.
[11] Ostrowski T. (2009). Cooperation of human as a strategy leading to success. The art of human resource management (Chapter 2). Ed. G. Drozdowski, Z. Dobrowolski, The State Vocational University in Gorzów Wlkp, Institute of Management, p. 27–36.
O BEZPIECZNYM WYBORZE MIESZANYM PRZY DECYZJACH W WARUNKACH NIEPEWNOCI
Streszczenie
W pracy zajmujemy siĊ bezpiecznym wyborem mieszanym, zwanym takĪe wybo-rem mieszanym o zerowym ryzyku, w pewnych problemach decyzyjnych podejmowanych w warunkach niepewnoĞci. Podane są warunki konieczne i dosta-teczne istnienia tego typu wyboru, który w jĊzyku teorii gier okreĞlany jest jako równowaga mieszana Nasha w grze symetrycznej. Zarówno warunki istnienia mie-szanego wyboru o zerowym ryzyku, jak i wzory na jego wyznaczanie są wyraĪone przy pomocy macierzy punktu siodłowego.
Słowa kluczowe: wybór mieszany, równowaga mieszana Nasha
Tadeusz Ostrowski Instytut Techniczny
PaĔstwowa WyĪsza Szkoła Zawodowa w Gorzowie Wlkp. ul. MyĞliborska 34, 66-400 Gorzów Wlkp.
