GHG DATABASE
15 Appendix F: Detailed results of final evaluation
15.1.3 Improving the Platform
3.3.1 Extensive form of Game Representation: The extensive form can be used to formalize games with a time sequencing of moves whereby the node denotes a player’s choice. The player is specified by a number listed by vertex with promising move of a player symbolized by line out of the vertex.
3.3.2 Strategic form of Game Representation: The strategic form is the matrix form of game representation which shows players, strategies, and payoffs as shown in matrix below. As shown in table 2, 16 is the payoff received by the row player (player C) while 12 is the payoff for the column player (Player D).
Table 2: Payoff Matrix of 2-Player, 2-Strategy Game
Player D Player D
Player C 16, 12 -3,-3
Player C 0,0 -12, -16
SELF ASSESSEMENT EXERCISE
Discuss the different types of games, discuss their similarities and dissimilarities 3.4 Applications of Game Theory:
i. Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.
ii. Game theoretic analysis aid bargaining, oligopolies and mergers & acquisitions pricing, fair division, duopolies,
iii. Game theoretic analysis aid information economics and industrial organization, and political economy.
SELF ASSESSEMENT EXERCISE
Discuss the various areas of applications of game theory 3.4. Solving Numerical Problems of Game Theory
Numerical Example 1: Consider a market whose inverse demand function is given by:
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Q
p=60−2 60 2Q−
There are two firms in the market facing the Cournot type of competition. The cost functions of the firms are given by:
1 1
2
2 2
C 30
C 2
q q
=
=
(a) Find the firm whose marginal cost is relatively constant and the firm whose marginal cost is relatively rising
(b) Derive the reaction functions of both firms.
(c) What are the Cournot equilibrium quantities and price?
Solution to Numerical Example 1: Obtain the marginal cost function of both firms as follows:
Firm 1 marginal cost is relatively constant because,
1 1
dC 30 dq = Firm 2 marginal cost is relatively rising in q2 because,
1 2 2
dC 4 dq = q
The reaction functions are derived as follows:
1 1 1
1 1
1 2 1 1
2
1 1 2
1
1 2
1
1 1
1 2
1 2
2 1
( ) ( )
(60 2 ) 30
[60 2( )] 30
60 2 2 30
60 4 2 30
0
60 4 2 30 0
4 30 2
30 2 4
P Q q c q
Q q q
q q q q
q q q
q q
q Setting
q
q q
q q
q q
= −
= − −
= − + −
= − − −
= − − −
=
− − − =
= −
= −
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2 2 2
2
2 2
2
1 2 2 2
2 2
1 2 2 2
2
1 2 2
2
( ) ( )
(60 2 ) 2
[60 2( )] 2
60 2 2
60 2 2 4
P Q q c q
Q q q
q q q q
q q q q
q q q
q
= −
= − −
= − + −
= − − −
= − − −
2 2
1 2 2
1 2
1 2
0
60 2 2 4 0
60 2 6 0
60 2 6 Setting
q
q q q
q q
q q
=
− − − =
− − =
= −
the reaction functions of both firms are thus given by: 1 30 2 2 4
q = − q & 2 60 2 1 6
q = − q respectively.
Substituting the RF q1 into RF q2, we have:
1 2
1 2
1 2
1 2
2 2
30 4 2 0
60 2 6 0 ( )
30 4 2 0
120 4 12 0
90 10 0
9
q q
q q by elimination
q q
q q
q q
− − =
− − =
− − =
− + + =
− + =
=
1 2
1
1
* * *
*
30 2(9) 4 3
( ) 12
60 2 60 2(12)
36 q
q
Q q q
P Q
P
= −
=
= + =
= −
= −
=
The Cournot equilibrium quantities and price are, q1=3, q2 =9&P=36 SELF ASSESSMENT EXERCISE
Describe the relationship between an explicit and an implicit function.
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4.0 CONCLUSION
Game theory is a theory of tactical relations among rational decision-makers. The application of game theory is all encompassing in social sciences. When a game is presented in normal form, it is presumed that each player acts simultaneously, that is, without knowing the move/actions of the other player. On the other hand, extensive form representation shows players have prior knowledge about each other’s strategy.
5.0 SUMMARY
In this unit, we have discussed the game vs. game theory, explain the different types of games, provided extensive form of game representation, strategic form of game representation.
6.0 TUTOR-MARKED ASSIGNMENT 1. Consider the following pay-off matrix:
Player I Player II
A B C
A 12 20 32
B 12 4 56
C 4 12 -3
Determine the strategy that each of the two players should play.
2. Consider the following pay-off matrix of a new contract for academic staff whereby university management and ASU entered into negotiation that resulted in ASU making 3 different proposals and management making 3 different proposals.
ASU Proposals
Management Proposals
G Q W
V 19 24 14
S 14 15.6 12.5
U 12 17 18
(a) Determine if there is an agreement between ASU and Management. Hint, find the point of equilibrium if there is
(b) Calculate the mixed strategies for management and ASU (c) What is the optimum strategy for ASU?
(d) What is the optimum strategy for management?
3. Consider a game between propsective university students namely, Yinka (Y) and her father Momodu (M) such that Yinka A has to choose whether to seek admission into a univefrsity that cost N2,000 per semester or not. Momodu has to decide whether to pay get Yinka educated with a fee of N20,000 or both Yinka and her father share the family income equally. Yinka’s
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education and family income sharing has some impact on family’s productivity. With no education and sharing of family’s income, the productivity of the family would be N30,000, while if either education or family income is shared the family’s productivity would rise and become N40,000. If both education and family income is shared, the producitity of the family would be N48,000.
(a) Construct the pay-off matrix for the game (b) Is there any equilibrium in dominant strategies?
(c) Can you find the solution of the game with Iterated Elimination of Dominated Strategies?
(d) Is there any Nash equilibrium?
7.0. REFRENCES/FURTHER READING
Aumann, Robert; & Hart, Sergiu (1992). "Handbook of Game Theory with Economic Applications". 1: 1 - 33.
Aumann, Robert J.; & Heifetz, Aviad (2002). Handbook of Game Theory with Economic Applications. pp. 1665 - 1686.
Bicchieri, Cristina (1989), "Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge", Erkenntnis 30 (1-2): 69 - 85.
Camerer, Colin (2003). "Introduction", Behavioral Game Theory: Experiments in Strategic Interaction, Russell Sage Foundation, pp. 1 - 25.
Carl Shapiro (1989). "The Theory of Business Strategy," RAND Journal of Economics, 20(1), pp. 125 - 137.
Chastain, E. (2014). "Algorithms, Games, and Evolution". Proceedings of the National Academy of Sciences, 111 (29): 10620 - 10623
Colin F. Camerer (1997). "Progress in Behavioral Game Theory," Journal of Economic Perspectives, 11(4), pp. 167 - 188.
Colin F. Camerer (2003). Behavioral Game Theory, Princeton.
Colin F. Camerer George Loewenstein, & Matthew Rabin, ed. (2003). Advances in Behavioral Economics, Princeton.
Drew Fudenberg (2006). "Advancing Beyond Advances in Behavioral Economics," Journal of Economic Literature, 44(3), pp. 694 - 711.
Dutta, Prajit K. (1999). Strategies and games: theory and practice. MIT Press.
Eric Rasmusen (2007). Games and Information, (4th ed.), Description.
Fernandez, L F.; Bierman, H S. (1998), Game Theory with Economic Applications. Addison-Wesley.
Gibbons, Robert (2001). A Primer in Game Theory. London: Harvester Wheatsheaf.
Gintis, Herbert (2000). Game Theory Evolving: A Problem-Centered Introduction to Modelling Strategic Behavior, Princeton University Press.