University of Hong Kong
ECON6021
Chiu
Game Theory
The essence of a game of strategy is the interdependence of the player’s decisions.
These interactions arise in two ways. The first is sequential in which players make
alternating moves. Each player, when it is his turn, must look ahead to how his current
actions will affect the future actions of others, and his own future actions in turn.
The second kind of interaction is simultaneous in which players act at the same time,
in ignorance of the others’ current actions. However, each must be aware that there are other
active players, who in turn are similarly aware, and so on.
When you find yourself playing a strategic game, you must determine whether the
interaction is simultaneous or sequential. Some games such as football have elements of both.
A game in general consists of three elements: (1) players; (2) a set of strategies for
each player; and (3) a payoff function for each player. All of the above information is
commonly known among the players. An equilibrium is a complete description/prescription
of actions chosen for all players in such a way that given those prescribed actions by other
players, no player will have a unilateral incentive to deviate from his/her prescribed actions.
This notion of equilibrium was defined and was proved to exist for a great class of games by
John Nash in 1950. Therefore, such an equilibrium concept is commonly known as Nash
equilibrium. John Nash received the Nobel Prize in economics together with two other game
theorists in 1995.
Simultaneous Games
Game 1: Over-Cover Warfare
Every week, Next magazine and Oriental weekly compete to have the most eyecatching cover story. A dramatic or interesting cover will attract the attention of potential
buyers at newsstands. Thus every week the editors of Next magazine meet behind closed
doors to select their cover story. They do so with the knowledge that the editors of Oriental
are meeting elsewhere, also behind closed doors, to select their cover. The editors of Oriental
in turn know that the editors of Next are making a similar decision, those of Next know that
those of Oriental know and so on.
In the competition between Next and Oriental, think of a hypothetical week that
produces two major news stories: the 911 story and HK’s economic situation. The editors’
choice of cover story is primarily based on what will attract the most newsstand buyers
(subscribers buy the magazine whatever the cover is). Of these newsstand buyers, suppose 70
percent are interested in the impeachment of 911 story and 30 percent in HK’s economic
situation. These people will buy the magazine only if the story that interests them appears on
the cover, if both magazines have the same story, the group interested in it splits equally
between them. What will the two magazines’ decisions?
Next
911
HK
Oriental
911
35, 35
30, 70
HK
70, 30
15, 15
1
A strategy is a dominant strategy for a player if that strategy gives a higher payoff
than other strategies of his do, irrespective of the other player’s strategy. Games in which
each side has a dominant strategy are the simplest games from the strategic perspective.
There is strategic interaction, but with a foregone conclusion. The Nash equilibrium where
both parties adopt their dominant strategies is called dominant strategy equilibrium. Note
that all dominant strategy equilibria must be Nash equilibria, but not vice versa.
Sometimes one player has a dominant strategy but the other does not. Games in
which only one side has a dominant strategy are also very simple. Why? We can sum up the
lessons of these examples into a rule for behavior in games with simultaneous moves.
Rule 1: If you have a dominant strategy, use it.
Rule 2: Eliminate any dominated strategies from consideration, and go on doing so
successively.
Game 2: Prisoners’ Dilemma
Two suspects were prosecuted for robbing a bank. Now they are under investigation
separately and simultaneously. What will each of these suspects do to minimize his year of
imprisonment?
suspect 2
suspect 1
confess
confess
10,10
not to confess
0,20
not to confess
20,0
1,1
Note: the number in each cell indicates the year of imprisonment.
Game 2a: Another version of Prisoners’ Dilemma
Tchaikovsky
composer
confess
confess
10,10
not to confess
0,20
not to confess
20,0
1,1
Game 3: Warfare between the mainland and Taiwan (purely fictitious)
A
D
G
B
E
H
C
F
I
2
The grid above shows the positions and the choices of the combatants. A Taiwanese
ship at the point I is about to fire a missile, intending to hit a Chinese ship at A. The missile’s
path is programmed to hit at the launch; it can travel in a straight line, or make sharp right
angled turns every 20 seconds. If the Taiwanese missile flew in a straight line from I to A,
Chinese missile defenses could counter such a trajectory very easily. Therefore, the
Taiwanese will try a path with some zigzags. All such paths that can reach A from I lie along
the grid shown. Each length like IF equals the distance the missile can travel in 20 seconds.
The Chinese ship’s radar will detect the launch of the incoming Taiwanese missile,
and the computer will instantly launch an antimissile. The antimissile travels at the same
speed as the Taiwanese missile, and can make similar 90-degree turns. So the anti-missile’s
path can also be set along the same grid starting at A. However, to allow for enough
explosives to ensure a damaging open-air blast, the antimissile has only enough fuel to last
one minute, so it can travel just three segments (e.g. A to B, B to C, and C to F, which we
write as ABCF).
If, before or at the end of the minute, one antimissile meets the incoming missile, it
will explode and neutralize the threat. Otherwise the missile will go on to hit the Chinese
ship. The question is: How should the trajectories of the two missiles be chosen?
C1-ABCF
C2-ABEF
C3ABEH
C4ABED
C5ADGH
C6ADEH
C7-ADEF
C8ADEB
T1IFCB
H
O
O
T2IFEB
O
H
H
T3IFED
O
H
H
T4IFEH
O
H
H
T5IHGD
O
O
O
T6IHED
O
H
H
T7IHEB
O
H
H
T8IHEF
H
H
H
O
H
H
H
H
H
H
H
O
O
O
H
H
O
O
O
O
H
H
H
O
H
H
H
O
H
H
H
H
H
H
H
O
O
H
H
H
H
H
H
Equilibrium strategies
When all simplifications based on dominant and dominated strategies have been used, the
game is at its irreducible minimum level of complexity and the problem of the circular
reasoning must be confronted head-on. What is best for you depends on what is best for your
opponent and vice versa. Here we introduce the concept of equilibrium, or Nash equilibrium
in honor of John Nash who developed the concept. An equilibrium is a combination of
strategies in which each player’s action is the best response to that of the other. Given what
the other is doing, neither wants to change his own move.
Rule 3: Having exhausted the simple avenues of looking for dominant strategies or
ruling out dominated ones, the next thing to do is to look for an equilibrium of the game.
Remark: For some games, a pure strategy equilibrium may not exist (e.g., the game of
paper-rock-scissors). In that case, we have to appeal to mixed strategies, for almost all
games, a mixed strategy equilibrium always exists.
3
Game 4: Collusion among players of unequal size
An important feature of OPEC is that its members are of unequal size. Saudi Arabia is
potentially a much larger producer than any of the others. Do large and small members of a
cartel have different incentives to cheat?
We keep matters simple by looking at just one small country, say Kuwait. Suppose that
in a cooperative condition, Kuwait would produce 1 million barrels per day, and Saudi Arabia
would produce 4. For each, cheating means producing 1 million extra barrels a day. So
Kuwait's choices are 1 and 2; Saudi Arabia's, 4 and 5. Depending on the decisions, total output on the market can be 5, 6, or 7. Suppose the corresponding profit margins (price minus
production cost per barrel) would be $16, $12, and $8 respectively. This leads to the
following profit table. In each box, the bottom left number is the Saudi profit, and the top
right number is the Kuwaiti profit, each measured in millions of dollars per day.
Profits (Millions of Dollars/Day) for Saudi Arabia, Kuwait
Saudi
Arabia
Production
4
5
1
64, 16
60, 12
Kuwait production
2
48, 24
40, 16
Kuwait has a dominant strategy: cheat by producing 2. Saudi Arabia also has a dominant
strategy, but this is the cooperative output level of 4. The Saudis cooperate even though
Kuwait cheats. The prisoners' dilemma has vanished. Why?
Game 5: Output Decisions in Duopoly



there are two firms producing the same product
each firm can produce 1, 2, or 3 units of output at a zero marginal cost
the (inverse) market demand function is as follows
Quantity demanded
2
3
4
5
6


Price
16
10
6
3
1.5
How much should each firm produce to maximize their joint profits?
How much will each firm produce based on individual’s incentive?
Construct a payoff matrix to answer the above two questions.
Sequential Games
The general principle for sequential-move games is that each player should figure out
the other players’ future responses, and use them in calculating his own best current move.
So important is this idea that it is worth codifying into a basic rule of strategic behavior.
Rule 4: Look ahead and reason back.
4
Anticipate where your initial decisions will ultimately lead, and use this information
to calculate your best choice. Most strategic situations involve a longer sequence of decisions
with several alternatives at each, and mere verbal reasoning cannot keep track of them.
Successful application of the rule of looking ahead and reasoning back needs a better visual
aid. A “tree diagram” of the choices in the game is one such aid. We can get to HKU campus
from TST via different routes. We can show them schematically:
Admiralty
MTR station
take a bus
HKU
Take MTR
take a ferry
TST
take a minibus
HKU
Central
HKU
take a taxi
This road map, which describes one’s options at each junction, looks like a tree with
its successively emerging branches--hence the term “decision tree.” We can use such a tree
to depict the choices in a game of strategy, but one new element enters the picture. A game
has two or more players. At various branching points along the tree, it may be the turn of
different players to make the decision. A person making a choice at an earlier point must
look ahead, not just to his own future choices, but to those of others. To remind you of the
difference, we will call a tree showing the decision sequence in a game of strategy a game
tree, reserving the term decision tree for situations in which just one person is involved.
Game 6: Suppose the market for some good is dominated by an incumbent, and a new firm is
deciding whether to enter the market. We show the structure of moves and payoffs in the
following game tree:
Accommodate
$100,000 to New Firm
Enter
Incumbent
Fight Price War
-$200,000 to New Firm
New Firm
$0 to New Firm
Keep Out
5
What should the new firm do? It has to assign how likely the incumbent will
accommodate or make a war. Such probabilities come from the new firm’s beliefs about the
incumbent’s profits in each of these cases. Such information is added to the game tree. What
will be the outcome?
Accommodate
$100,000 to New Firm
$100,000 to Incumbent
Enter
Incumbent
Fight Price War
New Firm
Keep Out
-$200,000 to New Firm
-$100,000 to Incumbent
$0 to New Firm
$300,000 to Incumbent
Traditionally, a sequential game is represented as a normal (more correctly, strategic) form. A
strategy of a player thus is a complete description of what a player will do under whatever
conceivable scenarios of the game. The above sequential game is thus represented by the
following normal form. Note that there are two Nash equilibria in this normal form game:
(Enter, accommodate) and (keep out, fight price war). However, some thought tells us that
giving that new firm has already entered, it is not in the interest of the incumbent to fight
price war because, given that situation, not fighting price war is more profitable than fighting.
Hence, we should rule out (keep out, fight price war) as an equilibrium because fighting price
war is not a credible threat.
incumbent
accommodate
enter
fight price war
(100k,100k)
(-200k,-100k)
(0,300k)
(0,300k)
new firm
keep out
6
Remark: For sequential games, an equilibrium consists of instructions regarding players’
prescribed actions even for the point which is never reached should the equilibrium strategies
are followed. It is reasonable to impose a condition for an equilibrium: even at points where
are never reached the agents at those points are still required to act optimally even if the
equilibrium does not prescribe the point to be reached at positive possibility. Equilibria
survive such a refinement is called subgame perfect. Subgame perfect equilibria rule out
incredible threats.
Game 7: Judicial Procedures (Difficult !!!)
There are three alternative procedures to determine the outcome of a criminal court
case. Each has its merits, and you might want to choose among them based on some
underlying principles.
1. Status Quo: First determine innocence or guilt, then if guilty consider the appropriate
punishment.
2. Roman Tradition: After hearing the evidence, start with the most serious punishment and
work down the list. First decide if the death penalty should be imposed for this case. If not,
then decide whether a life sentence is justified. If, after proceeding down the list, no sentence
is imposed, then the defendant is acquitted.
3. Mandatory Sentencing: First specify the sentence for the crime. Then determine whether
the defendant should be convicted.
The difference between these systems is only one of agenda: what gets decided first.
To illustrate how important this can be, we consider a case with only three possible outcomes:
the death penalty, life imprisonment, and acquittal.
The defendant’s fate rests in the hands of three judges. Their decision is determined
by a majority vote. This is particularly useful since the three judges are deeply divided.
Judge A holds that the defendant is guilty and should be given the maximum possible
sentence. This judge seeks to impose the death penalty. Life imprisonment is her second
choice and acquittal is her worst outcome.
Judge B also believes that the defendant is guilty. However, this judge unyieldingly
opposes the death penalty. Her most preferred outcome is life imprisonment. The precedent
of imposing a death sentence is sufficiently troublesome that she would prefer to see the
defendant acquitted rather than executed y the state.
Judge C holds that the defendant is innocent, and thus seeks acquittal. She is on the
other side of the fence from the second judge, believing that life in prison is a fate worse than
death. (On this the defendant concurs.) Consequently, if acquittal fails, her second-best
outcome would be to see the defendant sentenced to death. Life in prison would be the worst
outcome.
7
Best
Middle
Worst
Judge A’s ranking
Death Sentence
Life in Prison
Acquittal
Judge B’s ranking
Life in Prison
Acquittal
Death Sentence
Judge C’s ranking
Acquittal
Death Sentence
Life in Prison
The difference between these systems is only one of agenda: what gets decided first. To
illustrate how important this can be, we consider a case with only three possible outcomes: the
death penalty, life imprisonment, and acquittal.
The defendant’s fate rests in the hands of three judges. Their decision is determined
by a majority vote. This is particularly useful since the three judges are deeply divided.
What will the judges do under different procedures?
8