The Influence of Nature on Outcomes of Three Players Game
Dr. Sea-shon Chen, Department of Business Administration, Dahan Institute of Technology, Taiwan
ABSTRACT
This paper based on Nash equilibrium theory to explore the uncertainty of Nature that influences
the outcome of the three player’s game. Nature has two phases; one is exogenous uncertainty or random
and another is endogenous uncertainty or strategy choosing. Exogenous random is the environment or
situation unpredictable. Endogenous uncertainty is the players’ unpredictable intentions that result in
mixed strategy in the game. The paper expresses the methodology and utilize computer to explore
discrete equilibrium of the game, then connect discrete equilibriums into continuous results. The result
reveals that pure equilibriums and mixed equilibriums exist in different uncertainty range. The results
also suggest, in the real world, the players know the information of Nature is important because that will
influence decision making and payoffs.
Keywords: Nature, exogenous uncertainty, endogenous uncertainty, Nash equilibrium
INTRODUCTION
Game theory is a modeling tool. It is an interdisciplinary and distinct approach to the study of
human behavior and strategic management (Perea, et al., 2006; Rasmusen, 1995; Saloner, 1991). The
disciplines most involved in game theory are mathematics, economics, social science, and behavioral
science. The essential elements of a game are (1) players, (2) actions, (3) information, (4) strategies, (5)
payoffs, (6) outcomes, and (7) equilibriums. Players are the individuals who make decision based on
that they are absolutely rational in their economic choices. Game theory is based on the assumption that
human beings are absolutely rational in their economic choices (von Neumann and Morgenstern, 2004).
Specifically, the assumption is that each person tries to maximize her or his rewards (utilities, profits,
incomes, or subjective benefits) in the circumstances that a player faces (Chen, 2008). This hypothesis
serves a double purpose in the study of the allocation of resources. First, it narrows the range of
possibilities; somewhat absolutely rational behavior is more predictable than irrational behavior. Second,
it provides a criterion for evaluation of the efficiency of an economic system. Sometimes the irrational
pseudo-player, Nature, takes random actions at specified points in the game with specified probabilities
(Rasmusen, 1995). Nature is indifferent to the outcomes, but the strategic players care about the
outcomes. Although the strategic players choose the strategies and the choices affect the outcomes, the
state of the world chosen randomly by Nature who is either exogenous or endogenous uncertain
probability and this information revealing basically affects the outcomes (Antonio, 2006; Bierman and
Fernandez, 1998; Shmaya, 2006). The value of information about Nature is very important; players may
or may not know the probability Nature will choose (Ponssard, 1976).
In many cases, the result we predict immensely influences our decision. The misplay of decision
is not due to shortage and/or error of information or the logic reasoning, but the phase we think is unique
(Cruz & Simaan, 2000; Zhu, 2004). We always put our hands to the phase that we customized and
overlook the changeful and uncertain reality. Eventually, our conservative and sealed vision results in
faulty judgment and decision-making. One of the game theory players, Nature, does not commit the
misplay of decision. Nature plays the game in many phases. Uncertainty or random is the reality of the
player who is Nature. Exogenous uncertainty is that the strategic game players do not know the
irrational pseudo-player has what kind of the state of the world. For example, three firms are deciding
whether and how to drill wells into a spring water deposit that lies under their adjacent land tracts. The
three firms do not know for sure whether there is water under their land or not, i.e. uncertainty of the state
of the world or the phase of player Nature. Endogenous uncertainty is that the strategic game players
choose among their action randomly, but sometimes with reasoning. In the non-cooperative game, when
more than one player adopts a mixed strategy (a probability stated strategy) these players randomize
independently of each other (Prasad, 2003; Ravikumar, 1987). Independence means that knowledge of
the strategy chosen by one player provides no new information about the strategy that will be chosen by
any other player who has adopted a mixed strategy in continuous or discontinuous game (Bierman and
Fernandez, 1998; Reny, 1999). For example, TAI-water, CLA-water, and HIB-water firm may choose
‘don’t drill’, a ‘narrow’ well, or a ‘wide’ well on certain probability. The probabilities which the firms
make decisions are endogenous uncertainty or psychological Nature.
Some papers or books introduce or investigate exogenous and endogenous uncertainty in static
games which limited in a case or study two games separately. This paper studies two kinds of Nature,
exogenous and endogenous uncertainty, impact on the equilibrium of games with computer. By using
computer to study, the processes are easy and the results can be continuous and illustrated.
PRINCIPLE
Exogenous and endogenous uncertainty in static games may result in mixed strategy. In game
theory a mixed strategy is a strategy which chooses randomly among possible moves. The strategy has
some probability distribution which corresponds to how frequently each move is chosen. A totally
mixed strategy is a mixed strategy in which the player assigns strictly positive probability to every pure
strategy. A mixed strategy should be understood in contrast to a pure strategy where a player plays a
single strategy with probability. Pure strategy Nash equilibriums are Nash equilibriums where all
players are playing pure strategies. Mixed strategy Nash equilibriums are equilibriums where at least
one player is playing a mixed strategy. During the 1980s, the concept of mixed strategies came under
heavy fire for being intuitively problematic. Randomization, central in mixed strategies, lacks
behavioral support (Shachat, and Swarthout, 2004). Seldom do people make their choices following a
lottery. This behavioral problem is compounded by the cognitive difficulty that people are unable to
generate random outcomes without the aid of a random or pseudo-random generator (Aumann, 1985).
However, rational strategies exist for finite normal form games under the assumption that strategy choices
can be described as choices among lotteries where players have security- and potential level preferences
over lotteries (Zimper, 2007).
Game theorist Rubinstein (1991) points out two alternative ways of understanding the concept: one
is to imagine that the game players stand for a large population of agents. Each of the agents chooses a
pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed
strategy hence represents the distribution of pure strategies chosen by each population. However, this
does not provide any justification for the case when players are individual agents. The other, called
purification is to suppose that the mixed strategies interpretation merely reflects our lack of knowledge of
the agent's information and decision-making process. Apparently random choices are then seen as
consequences of non-specified, payoff-irrelevant exogenous factors. However, it is unsatisfying to have
results that hang on unspecified factors, and this dismisses the possibility of a mixed-strategies analysis to
have any predictive power. Arguing that those factors are simply other players' beliefs about a player's
strategy, hence adopting a mixed strategy is the best response to a player playing mixed strategies, gives a
credible interpretation, but does not restore predictive power to the concept of mixed equilibriums.
Although economist’s attitude towards mixed strategies-based results has been ambivalent, mixed
strategies are still widely used for their capacity to provide Nash equilibrium in any game concerning
minimum monetary regret or maximum profits (Bade, 2005; Chong and Benli, 2005). The following is an
example.
Let (S, f) be a game, where Si is the strategy set for player i, S = S1 × S2 … × Sn is the set of strategy
profiles and f = (f1(x), ..., fn(x)) is the payoff function. Let x−i be a strategy profile of all players except for
player i. When each player i belongs to {1, ..., n} chooses strategy xi resulting in strategy profile x =
(x1, ..., xn), then player i obtains payoff fi(x). The payoff depends on the strategy profile chosen, i.e. on
the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy
profile x* belongs to S is a Nash equilibrium if no unilateral deviation in strategy by any single player is
profitable for that player, that is
i, xi  Si , xi  xi* : f i ( xi* , x*i )  f i ( xi , x*i ).
(1)
In order to investigate mixed strategies, noting pi is the payoff converted probability and p =
(p1(x), ..., pn(x)) is the function of the payoff converted probability. A strategy profile x* belongs to S is
a mixed equilibrium if no unilateral deviation in strategy by any single player has the most probable of
the payoff converted probability for that player, that is
i, xi  Si , xi  xi* : pi ( xi* , x*i )  pi ( xi , x*i ).
(2)
A game can have a pure strategy Nash equilibrium (NE) or a mixed NE in its mixed extension that
of choosing a pure strategy stochastically with a fixed frequency. Nash explained that, if we allow
mixed strategies, i.e. players choose strategies randomly according to the most probable probabilities then
every n-player game in which every player can choose from finitely many strategies admits at least one
Nash equilibrium.
METHODOLOGY
A good model in game theory has to be realistic in the sense that it provides the perception of real
life social phenomena (Rubinstein, 1991). The paper based on this argument and the principles to
investigate pure strategy and possible mixed strategy. A situation is three firms TAI, CLA, and HIB are
deciding whether and how to drill wells into a mineral water resources that lie under their adjacent tracts.
Because the three firms do not know for sure whether there is mineral water under their tracts or not, the
consequence of their actions depends on Nature (exogenous uncertainty) that beyond their knowledge and
control. That is the outcome of the game depends on the state of the world chosen randomly by Nature
(the state of the world) and the strategies chosen by the strategic players (TAI, CLA, and HIB). Suppose
there possible states of the world: either there is a deposit of x billion gallon water under the land and any
well will be a gusher (probability p), or there is no water under the land and any well will be a dry hole
(probability q or 1 – p). The two states of the world (Nature) and the possible strategy profiles result in
54 possible outcomes. A three dimensional matrix (3×3×3) is suggested to represent the payoff matrix of
the mineral water drilling game when Nature is gushing (probability p). For example, M132 = (pT132,
p
C132, pH132) means the profit for each firm if TAI chooses don’t drill, CLA chooses wide, and HIB
chooses narrow; M213 = (pT213, pC213, pH213) means the profit that TAI chooses narrow, CLA chooses don’t
drill, and HIB chooses wide. Another matrix (3×3×3) represents the payoff matrix of the game when
Nature is dry well (probability q). For example, N131 = (qT131, qC131, qH131) means the profit for each firm if
TAI chooses don’t drill, CLA chooses wide, and HIB chooses don’t drill; N212 = (qT212, qC212, qH212)
means the profit that TAI chooses narrow, CLA chooses don’t drill, and HIB chooses narrow. The
expected payoff matrix Gijk for the mineral water drilling game is suggested. For example, in ((3×3×3)
expected payoff matrix G312 = (Tu312, Cu312, Hu312), the elements will be counted as Tu312 = pT312 × p + qT312
× q, Cu312 = pC312 × p + qC312 × q, and Hu312 = pH312× p + qH312 × q.
Pure Strategy
By using Excel, if input data into matrixes Mijk and Nijk the resulting Gijk will provide information
for equilibrium judgment or advanced mixed strategy measurement. An empirical case study with digits
will perform in the following. Table 1 and Table 2 are the data of payoff matrix of the mineral water
drilling game if gashing and if dry well, respectively. Table 3 is the format of expected payoff matrix of
the mineral water drilling game.
Table 1: The Payoff Matrix of the Mineral Water Drilling Game if Gushing
Nature: Gushing (probability p)
HIB (Don’t drill)
TAI
Don’t drill
Narrow
Wide
HIB (Narrow)
TAI
Don’t drill
Narrow
Wide
HIB (Wide)
TAI
Don’t drill
Narrow
Wide
Don’t drill
0
0
45
0
30
0
0
0
0
Don’t drill
0
0
40
50
0
0
30
0
0
Don’t drill
0
0
30
45
-5
0
30
0
15
0
15
10
CLA
Narrow
40
30
10
0
0
0
0
30
5
Wide
30
20
10
0
0
0
0
15
20
CLA
Narrow
50
30
-5
20
20
-10
0
-5
5
Wide
30
20
5
25
12
10
0
15
18
CLA
Narrow
30
15
-2
0
2
0
0
-2
5
Wide
30
18
5
20
1
3
Mixed Strategy
If there is no pure strategy equilibrium, mixed strategy equilibrium will be figured out by the
expected payoff and probabilities. Let pTD, pTN, and pTW denoted the probability of TAI choosing Don’t
drill, Narrow, and Wide. Let pCD, pCN, and pCW denoted the probability of CLA choosing Don’t drill,
Narrow, and Wide. And, let pHD, pHN, and pHW denoted the probability of HIB choosing Don’t drill,
Narrow, and Wide, respectively. The mixed strategy profile will be figured out by comparing these
probabilities.
The process to find the most possible probability set is first computing the average and standard
deviation of all the elements to get two 3 × 3 matrixes; one is the average and the other is the standard
deviation. For example, referring to Table 3, the average of Tu1jk (j and k = 1, 2, 3) is the element of
average matrix a11 and the standard deviation of Hu2jk (j and k = 1, 2, 3) is the element of standard
deviation s23. Second, the standard normal random variable, Z, is calculated. Third, the probability of each
Z is calculated. Finally, comparing the firm probability, the combination of probability for mixed strategy
will be found.
Table 2: The Payoff Matrix of the Mineral Water Drilling Game if Dry Well
Nature: Dry well (probability 1 − p)
HIB (Don’t drill)
TAI
Don’t drill
0
0
-15
-5
-30
0
Don’t drill
Narrow
Wide
HIB (Narrow)
TAI
0
-15
-30
0
0
-5
0
-15
-30
Wide
-30
-20
-10
0
0
-10
0
-15
-30
CLA
Narrow
-20
-10
-5
-25
-10
-5
0
-15
-30
Wide
-30
-20
-10
-20
-5
0
0
-15
-15
CLA
Narrow
-20
-15
-2
0
2
-30
0
-2
-20
Wide
0
-16
-30
-15
-3
-15
Don’t drill
0
0
0
-15
-10
-20
-30
0
-10
Don’t drill
Narrow
Wide
HIB (Wide)
TAI
0
0
0
CLA
Narrow
-20
-10
-5
Don’t drill
0
0
-30
-15
-12
-5
-30
-5
-15
Don’t drill
Narrow
Wide
Table 3: The Expected Payoff Matrix of the Mineral Water Drilling Game
CLA
HIB (Don’t drill)
Don’t drill
Narrow
Wide
T
C
H
T
C
H
T
C
Don’t drill
u111
u111
u111
u121
u121
u121
u131
u131
T
C
H
T
C
H
T
C
TAI
Narrow
u211
u211
u211
u221
u221
u221
u231
u231
T
C
H
T
C
H
T
C
Wide
u311
u311
u311
u321
u321
u321
u331
u331
HIB (Narrow)
TAI
Don’t drill
Narrow
Wide
T
u112
T
u212
T
u312
HIB (Wide)
TAI
Don’t drill
Narrow
Wide
T
u113
T
u213
T
u313
Don’t drill
C
H
u112
u112
C
H
u212
u212
C
H
u312
u312
Don’t drill
C
H
u113
u113
C
H
u213
u213
C
H
u313
u313
T
u122
T
u222
T
u322
T
u123
T
u223
T
u323
CLA
Narrow
C
u122
C
u222
C
u322
CLA
Narrow
C
u123
C
u223
C
u323
H
u122
H
u222
H
u322
T
H
T
u123
H
u223
H
u323
u132
T
u232
T
u332
u133
T
u233
T
u333
Wide
C
u132
C
u232
C
u332
Wide
C
u133
C
u233
C
u333
H
u131
u231
H
u331
H
H
u132
u232
H
u332
H
H
u133
u233
H
u333
H
RESULTS
Based on the principle and methodology, the study assumes asymmetrical date matrix (Table 1 and
Table 2) and uses computer Excel to explore the pure strategy equilibrium and mixed strategy
equilibriums. The results are also explained by three tables (Tables 4, 5, and 6) to express the pure and
mixed equilibrium.
Pure Strategies
An example of asymmetrical data matrixes is suggested. After analysis the data with exogenous
uncertainty (gushing probability p = 0.90, dry well probability q = 0.10) and the expected payoff matrix
are listed in Table 4. The unique strategy for Nash (pure) equilibrium of the game is {Wide, Wide,
Narrow} because of Tu332 (1.5) > Tu132 (0) > Tu232 (-6), Cu333 (3.5) > Cu313 (0) > Cu323 (-5), and Hu332 (9) >
H
u333 (1.2) > Hu331 (-1).
Table 4: The Expected Payoff Matrix of the Mineral Water Drilling Game (p = 0.90)
CLA
HIB (Don’t drill)
Don’t drill
Narrow
Wide
Don’t drill
0
0
0
0
34
0
0
24
TAI
Narrow
39
-0.5
0
12
26
0
25.5
16
Wide
24
0
0
6
8.5
-0.5
1.5
8
HIB (Narrow)
TAI
Don’t drill
Narrow
Wide
HIB (Wide)
TAI
Don’t drill
Narrow
Wide
Don’t drill
0
0
36
43.5
-1
-2
24
0
-1
Don’t drill
0
0
24
39
-5.7
-0.5
24
-0.5
12
0
0
-1
0
12
15
CLA
Narrow
43
26
-5
15.5
17
-9.5
0
-6
1.5
Wide
24
16
3.5
20.5
10.3
9
0
12
14.7
CLA
Narrow
25
12
-2
0
2
-3
0
-2
2.5
Wide
27
14.6
1.5
16.5
0.6
1.2
If the range of gushing probability, p, is 1.0 ≥ p ≥ 0.8572, The expected payoff functions for
TAI-water (uT), CLA-water (uC), and HIB-water (uH) are Equation 3, 4, and 5.
TAI:
uT = 35.008 p − 30.0071,
(3)
CLA:
uC = 14.992 p − 9.9926,
(4)
and
HIB:
uH = 10.008 p − 0.0074.
(5)
Using the former stated rule, for the range of p is 0.75 ≥ p ≥ 0.50, the unique strategy for Nash (pure)
equilibrium of the game is {Narrow, Narrow, Narrow}, and the expected payoff functions are Equation 6,
7, and 8.
TAI:
uT = 30 p − 15,
(6)
CLA:
uC = 30 p − 10,
(7)
and
HIB:
uH = 40 p − 10.
(8)
For the range of p is 0.49 ≥ p ≥ 0.4286, the unique strategy for Nash (pure) equilibrium of the game
is {Don’t drill, Wide, Wide}, and the expected payoff functions are Equation 9, 10, and 11.
TAI:
uT = 0,
(9)
CLA:
uC = 29.683 p − 0.1504,
(10)
and
HIB:
uH = 35.008 p − 15.004.
(11)
For the range of p is 0.4285 ≥ p ≥ 0.3334, the unique strategy for Nash (pure) equilibrium of the
game is {Don’t drill, Narrow, Don’t drill}, and the expected payoff functions are Equation 12, 13, and 14.
TAI:
uT = 0,
(12)
CLA:
uC = 59.855 p − 19.953,
(13)
and
HIB:
uH = 0.
(14)
For the range of p is 0.2308 ≥ p ≥ 0.001, the unique strategy for Nash (pure) equilibrium of the
game is {Don’t drill, Don’t drill, Narrow}, and the expected payoff functions are Equation 15, 16, and 17.
If gushing probability, p = 0.00, the equilibrium strategy is {Don’t drill, Don’t drill, Don’t drill}.
TAI:
uT = 0,
(15)
CLA:
uC = 0,
(16)
and
HIB:
uH = 39.994 p + 0.0003.
(17)
Mixed Strategy
If the gushing probability p is 0.857 ≥ p ≥ 0.751, then the pure strategy does not exist. For
example, if the situation is gushing probability p = 0.80 and dry well probability q = 0.20, the expected
probability matrix are listed in Table 5.
Table 5: The Expected Probability Matrix of the Mineral Water Drilling Game (p = 0.80)
CLA
HIB (Don’t drill)
Don’t drill
Narrow
Wide
Don’t drill
0.17
0.27
0.06
0.17
0.27
0.06
0.17
0.19
0.06
TAI
Narrow
0.29
0.06
0.12
0.11
0.30
0.12
0.21
0.21
0.12
Wide
0.29
0.15
0.18
0.08
0.32
0.15
0.04
0.31
0.13
HIB (Narrow)
TAI
Don’t drill
Narrow
Wide
HIB (Wide)
TAI
Don’t drill
Narrow
Wide
Don’t drill
0.17
0.04
0.33
0.30
0.05
0.04
0.29
0.15
0.13
CLA
Narrow
0.17
0.31
0.11
0.30
0.19
0.03
0.18
0.33
0.02
0.17
0.03
0.04
Wide
0.19
0.21
0.22
0.23
0.29
0.31
Don’t drill
0.17
0.04
0.25
0.29
0.02
0.09
0.29
0.11
0.32
CLA
Narrow
0.17
0.21
0.11
0.17
0.21
0.08
0.06
0.16
0.06
0.17
0.04
0.06
Wide
0.24
0.20
0.08
0.20
0.12
0.16
Under the condition of gushing probability p = 0.8, the strategy for mixed equilibrium of the game
is {Wide, Don’t drill, Wide}, because TAI (Wide: pTW = 0.29), CLA (Don’t drill: pCD = 0.11), and HIB
(Wide: pHW = 0.32) are the most possible combination of the probability matrix. If the gushing probability
is 0.857 ≥ p ≥ 0.751, the mixed strategy is also {Wide, Don’t drill, Wide}, and the expected probability
functions for TAI-water (pT), CLA-water (pC), and HIB-water (pH) are Equation 18, 19, and 20.
TAI:
pT = – 0.4616 p2 + 0.981 p + 0.1996,
(18)
CLA:
pC = 0.1147,
(19)
and
HIB:
pH = – 0.1911 p2 + 0.3988 p – 0.119.
(20)
If gushing probability is 0.333 ≥ p ≥ 0.230, the pure strategy does not exist. The mixed strategy is
{Don’t drill, Narrow, Narrow}. For example, if p = 0.3, the expected probability matrix are listed in
Table 6.
Table 6: The Expected Probability Matrix of the Mineral Water Drilling Game (p = 0.30)
CLA
HIB (Don’t drill)
Don’t drill
Narrow
Wide
Don’t drill
0.17
0.08
0.06
0.17
0.08
0.06
0.17
0.05
0.06
TAI
Narrow
0.11
0.06
0.12
0.06
0.12
0.12
0.08
0.08
0.12
Wide
0.09
0.15
0.18
0.04
0.20
0.16
0.03
0.18
0.14
HIB (Narrow)
TAI
Don’t drill
Narrow
Wide
HIB (Wide)
TAI
Don’t drill
Narrow
Wide
Don’t drill
0.17
0.04
0.16
0.11
0.05
0.06
0.09
0.15
0.14
CLA
Narrow
0.17
0.10
0.06
0.12
0.06
0.09
0.07
0.19
0.11
0.17
0.04
0.03
Wide
0.05
0.08
0.14
0.09
0.16
0.23
Don’t drill
0.17
0.04
0.08
0.11
0.04
0.10
0.09
0.12
0.20
CLA
Narrow
0.17
0.06
0.06
0.08
0.08
0.12
0.06
0.13
0.08
0.17
0.05
0.04
Wide
0.08
0.08
0.05
0.08
0.11
0.14
Table 6 showed that TAI (Don’t drill: pTD = 0.17), CLA (Narrow: pCN = 0.10), and HIB (Narrow:
pHN = 0.07) are the most probable situation combination for the gushing probability p = 0.30. Actually,
in the range (0.333 ≥ p ≥ 0.230), the expected probability functions are Equation 21, 22, and 23.
TAI:
pT = 0.50,
(21)
CLA:
pC = 0.9943 p2 + 0.842 p + 0.039,
(22)
2
and
HIB:
pH = 0.3588 p + 0.3467 p + 0.1071.
(23)
DISCUSSION AND CONCLUSION
The results reveal that Nature influences the outcome of the games and makes the results to be very
complex especially if the players are three or more than three (Daskalakis and Papadimitriou, 2005). In
the results, the pure equilibrium strategy profile (Wide, Wide, Narrow) exists if gushing probability is in
the range of 1.0 ≥ p ≥ 0.8572. The pure strategy does not exist if 0.857 ≥ p ≥ 0.751; but, the mixed
equilibrium strategy profile is (Wide, Don’t drill, Wide). The pure equilibrium strategy profile (Narrow,
Narrow, Narrow) exists if 0.75 ≥ p ≥ 0.50. If 0.49 ≥ p ≥ 0.4286, the pure equilibrium strategy profile is
(Don’t drill, Wide, Wide). If 0.4285 ≥ p ≥ 0.3334, the pure equilibrium strategy profile is (Don’t drill,
Narrow, Don’t drill). If 0.333 ≥ p ≥ 0.230, the pure strategy does not exist; but, the mixed strategy is
(Don’t drill, Narrow, Narrow). If 0.2308 ≥ p ≥ 0.001, the strategy profile for pure equilibrium of the
game is (Don’t drill, Don’t drill, Narrow). If gushing probability p = 0.00, the equilibrium strategy
profile is (Don’t drill, Don’t drill, Don’t drill). To sum up, Table 7 lists all the situations.
Table 7: The Gushing Probability Range and Equilibrium Strategy Profile with Type
Gushing probability
Strategy profile (TAI, CLA, HIB)
Equilibrium profile
1.0 ≥ p ≥ 0.8572
(Wide, Wide, Narrow)
Pure
0.857 ≥ p ≥ 0.751
(Wide, Don’t drill, Wide)
Mixed
0.75 ≥ p ≥ 0.50
(Narrow, Narrow, Narrow)
Pure
0.49 ≥ p ≥ 0.4286
(Don’t drill, Wide, Wide)
Pure
0.4285 ≥ p ≥ 0.3334
(Don’t drill, Narrow, Don’t drill)
Pure
0.333 ≥ p ≥ 0.230
0.2308 ≥ p ≥ 0.001
p = 0.00
(Don’t drill, Narrow, Narrow)
(Don’t drill, Don’t drill, Narrow)
(Don’t drill, Don’t drill, Don’t drill)
Mixed
Pure
Pure
In the real world, Nature has two faces those are exogenous uncertainty and endogenous uncertainty.
Both of them will influence the process and results of games. The information is important, if the
probability of gashing is known, the firms may choose the best strategy to make profits. In the mixed
equilibrium situations, the endogenous uncertainty of decision making can be reduced by the gushing
information, too.
The special of this study is the resulting equilibrium expectation utilities or probabilities are
expressed in the form of equations, i.e. the solutions are segmental continuous. With the aid of
computer these results of games are easier to obtain. Further research may use the other software or
computer programming and suggest more complex situation to explore the game theory.
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