University of Hong Kong
ECON2101
Chiu
Additional materials on 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 eye-catching
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
1
HK
70, 30
15, 15
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 are 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
C
F
H
2
I
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 paperrock-scissors). In that case, we have to appeal to mixed strategies, for almost all games, a mixed
strategy equilibrium always exists.
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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.
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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
KCR 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
New Firm
$0 to New Firm
Keep Out
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-$200,000 to New Firm
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
Traditonally, an 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, accomodate)
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
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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. Son and mother: incredible commitment & moral hazard
To illustrate the importance of credibility of commitment, let us consider the following game between
a son and his mother.
Consider the following contract between a son and his mon to make sure that the son will go to sleep
at a right time. "The son goes to sleep by 10pm. In this case, he will receive no penalty. Otherwise, he
will be shot to death by his mon.'' Suppose the HK law allows this contract to be legal, and suppose
the pair arrange video recording of the son's activities so that the court can verify when he goes to
bed. Thus we have neither asymmetric information nor difficulty of verification. As long as the son
believes that the contract will be enforced, he will go to bed on time. So far so good. But will the mon
really want to kill the son when the latter does not abide by the contract? Definitely not. In case the
son doesn't go to bed by 10pm, it would be in both the mon and the son's interest to renegotiate the
contract. Because mon cannot commit to killing his son when the circumstance arises, the son knows
it and will simply ignore the contract. This commitment problem is also known as a time
consistency problem. The said contract is time inconsistent: At the outset, mon wants the court to
enforce the contract, but after the bad scenario occurs, she rather hopes that the court will not enforce
the contract. A time consistent contract is one in which under every contingency there does not exist
any alternative to the actions prescribed in contract that makes both parties better off (relative to
enforcement of the contract).
The above mentioned time inconsistency problem is also called moral hazard. Suppose a billionare
knows that his business partner in a southeast country is the son of the president of that country. Then
he may think that in case the business were unsuccessful, the president will come out to rescue the
firm. Hence, doing business with that business partner is a sure win, and this leads to excessive
investment and imprudential rescues. This whole sequence of irresponsible actions nowadays are
usually called moral hazard. (More will be said about it in the next topic.)
Game 8: to move or not to move
This form of moral hazard pointed out right before is closely related to government policy where
government is one party of a contract and citizens (or firms) are another party. I have recently read of
an interesting quote that illustrates this neatly: "There are two types of countries in the world: those
with deposit insurance, and those not knowing they have deposit insurance.'' The following game
illustrates the tension between a government and its citizens.
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Suppose residents near Yellow River are more vulnerable to attack of typhoons. Two states of nature:
good state and bad state with probability 0.9 and 0.1. In good state: loss to each resident near Yellow
River is zero, but in bad state, the loss is $1m. Residency elsewhere no loss under any state. Suppose
the moving cost = $50k , which is less than the expected loss of 0.1x$1m=100k. However, in case of
bad state, the government can step in to reduce the resident’s loss to $200k at a cost of $500k
Draw a game tree between a resident and a government and find the subgame prefect equilibrium of
it. (The resident's decision is to move or not to move. The government's decision is whether to rescue
if the resident has decided not to move and the bad state has occurred. Consider the corresponding
normal form game and find all Nash equilibria.
Game 9: Auctions (Optional, Omitted)
English auction---The auction begins with an opening bid. Given knowledge of the opening bid, the auctioneer
asks if anyone is willing to pay a higher price. The bids continue to rise until no other participants wish to
increase the bid. The highest bidder---the only bidder left---pays the auctioneer his or his bid and takes
possession of the item.
Second-price, sealed-bid auction---In such an auction, bidders submit bids without knowledge of the bids
submitted by others. The person submitting the highest bid wins, but has to pay only the amount bid by the
second highest bidder.
In both types of auctions, the winner will be the same person and has to pay the second highest valuation of the
machine. Show that bidding your true valuation of the item is your dominant strategy for second-price,
sealed-bid auction.
Game 10: 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
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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.
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?
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