# Sample Mathematics Paper on Analysis of Strategic Behavior in the Centipede game

Describe a unique subgame perfect Nash equilibrium of the centipede game (5 points)
The centipede game is a finite game between 2 players and is characterized by perfect
information. As the players alternate moves, each player is faced with an option of either
“continuing” or “stopping”. An option of “continue” has a reward of 3 which is added to the
total payoff of the player.This option also gives the other player a-go ahead to play. Ideally,
no player would choose to continue if he/she anticipates that the other player would stop in
the next period. 1 An option of “stop” ends the game and the payoffs have to be divided
between the players for the player who chooses to stop. The unique subgame Perfect
equilibrium of the centipede game is therefore the backward induction solution present for
each player to choose ‘stop’ at each stage. In other words, the unique subgame perfect
equilibrium is the extensive form of the perfect outcome of the subgame which prompts each
player to “quit” at each stage of the game irrespective of the periods in the centipede game.
Discuss why such Nash equilibrium appears counterintuitive (5 points);
While some games have multiple pure Nash equilibriums, others do not have any. Some of
these equilibriums are inconsistent with the intuitive notion of what the outcome of the game
should be and this brings about counterintuitive effects. 2 To illustrate further counterintuive
nature of the nash equilibrium, consider the following example:

Player 1 Player 2
B 1 B 2
Α 1 1,1 0,0
Α 2 0,0 0,0

The Nash equilibriums in this case are:
 A 1 B 1
 A 2 B 2

1. Kreps, David, Paul, M., John, R. &amp; Robert, W “Rational Cooperation: Repeated
Prisoner’s Dilemma,” Journal of Economic Theory, vol.2, no.7 (1982):245–252.
2. Von N. John, &amp; Oskar M. “Theory of Games and Economic Behavior,” (Princeton,
NJ, Princeton University Press,1944)

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The occurrence of two or more nash equilibriums as seen above is what causes of
counterintuitive. At these two equilibrium points, neither player can improve the payoffs by
unilaterally changing his/her strategy and this therefore prompts each player to play
rationally.
In the first equilibrium, the player 1 does much better than player 2, even though the two
players have exactly identical payoffs and strategies available (and conversely in the second
equilibrium).
The fact that no player would benefit by changing his strategy while the other players in a
pure nash equilibrium do not change their strategies explains further on the existence of
counterintuitive.

Use your practical understanding of strategic thinking to describe the most reasonable
prediction for the game (5 points).
Player 1 Player 2 Player 1 Player 2 Player 1 Player 2
R R R R R R

1
1




0
3




D

2
2




D

1
4




D D D D D
97 99 98
100 99 101

At the start of the game, each player receives a payoff of 1. After a series of many rounds,
player 1 and 2 finally reach the last round with each receiving a payoff of 100.
Since the centipede game is characterized with perfect information, each player knows best
the move of the other player. Hence, a player would be unwilling to proceed on with the
game in case he finds that he will lose upon playing. The most reasonable outcome for player
1 is 100 since it is the highest payoff. For player 2, the most reasonable outcome of the game
is the payoff is 101 since it is the highest payoff. However, this cannot be attained since each
player in the game would be unwilling to continue playing upon finding out that there is a
payoff loss to incur on continual play. Therefore, the most reasonable outcome of the game is
at the first entrance of the game where both player 1 and 2 receive a payoff of 1,1 each.

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Use your theoretical understanding of strategic thinking , discuss how such most
reasonable prediction can be rationalized/explained; DO NOT forget that both players
behave strategically in the game(15 points);-
To obtain an optimal behaviour with rationality in place, firms in a competitive market seek
to maintain price equal marginal cost given that they are profit maxi misers. “Though firms in
market scene fight to maximise profits, they are usually tamed by competition effects.” 3 This
leads to welfare gains to firms and customers. However, the resultant welfare differs as in the
case of a monopoly, since a non profit-maximizing monopolist can decided to be wasteful
and still make revenues. In actual sense, the nash equilibrium is never attained since firms
adjust to accommodate excess demand in some markets rather than pursuing the profit
maximization goal.
Additionally, take for example of a joint venture. Upon its formation, the venture will realize
a profit of \$X. If two countries with the joint venture agree to this pact, the total welfare will
be improved by an equivalent of N dollars. So the question now is, should the profits and
costs be split between the ventures? Well, such a query can be answered with two main
solutions and this are: the static bargaining approach by John Nash, as well as the bargaining
outcomes that form the perfect outcome of a strategic game.
The Nash solution puts forward the following notion. Imagine that there is a pie of size 1 to
divide. The threat point in this case will be: S 1 and S 2 . For pareto optimality, the best bargain
is the one that maximum.
(u 1 (p)-u 1 (S 1 ))*(u 2 (1-p)-u 2 (S 2 )),
Where;
u-utility.
p – share of the pie of size 1 that accrues to player 1.
S 1 – threat point for player 1.
S 2 -threat point for player 2.
3 .Von N. John, &amp; Oskar M. “Theory of Games and Economic Behavior,” (Princeton, NJ,
Princeton University Press, 1944)

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With a linear utility in place, if player 1 decides to leave, he will earn 0.3. On the other hand,
incase player 2 leaves; he will gain a payoff of 0. However, a total of 1 is earned by this joint
venture. The optimizing nash bargaining solution will be p that maximizes (p-.3)*(1-p-0);
that is, p=.65. The surplus that will be generate will be given by 1-.3-0=.7 and we each
player will get the outside portion + 1/2 the surplus. The difference in utilities can be
explained by factors such as risk aversion.
In the nash bargaining for a player, the joint ventures end up with a more worse utility
function. Despite both ventures joining hands to conduct business together and earn profits,
the patience of joint venture players is lost as each one of them starts to play strategically
each observing the action of the other. 4 When setting prices for example, one venture may set
a higher price while the other sets a low price in order to draw more demand. This makes the
ventures non-credible. As long as joint venture players prefer to be in equilibrium share their
revenues and costs together, each has to play rationally and strategically and be able to
dominate the

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Bibliography

1. Nash, J. F “Non-cooperative Games: Annals of Mathematics,” no.2 (1951): 286-295.
2. Harsanyi, John C., &amp; Reinhard, S, “A General Theory of Equilibrium: Selection in
Games,” (Cambridge, MA, MIT Press, 1998)
3. Kreps, David, Paul, M., John, R. &amp; Robert, W “Rational Cooperation: Repeated
Prisoner’s Dilemma,” Journal of Economic Theory, vol.2, no.7 (1982):245–252.
4. Von N. John, &amp; Oskar M. “Theory of Games and Economic Behavior,” (Princeton, NJ,
Princeton University Press,1944)
5. Shimoji, M. &amp; J. Watson, J “Conditional Dominance, Rationalizability, and Game
Forms,” Journal of Economic Theory, vol.8, no.3 (1998):161-195.