Lecture 1
Games in Extensive Form
TEACHING NOTES:
Main objective of Lecture 1 is to understand how games are written in tree form, and to able to solve
these games using backwards recursion. In the process, several common games will be introduced and
discusses to familiarize student with many basic concepts to be used throughout the lecture series.
Specific Concepts which should be understood, and Skills which the student should acquire:
By end of lecture, student should be able to:
1. Look at a tree diagram of a game and be able to understand and interpret it.
2. From a verbal description of a game, and write down the tree diagram for it.
3. Given a tree diagram, solve the game by backwards recursion.
4. Have some understanding of the conceptual basis (individualistic utility max) for the game
theoretic solution, and also some understanding of why it fails in real life games.
Specific Concepts which must be learned to acquire the skills listed above:
1. Elements of the Game Tree: Nodes, Branches, Payoffs.
2. Exercises in going from Game Tree to Verbal Description of Game and vice versa.
3. “Optimal” decisions for one player at one node. The Game Theoretic concept of maximizing an
individuals selfish payoff. Weaknesses when implemented in practice.
4. Using backwards recursion to solve games.
5. Solve specific games and discuss plausibility of solutions.
6. Describe empirical experience related to specific games, how it does not conform to game
theoretic predictions, and some discussion of why there is a conflict.
Game Theory
Game theory is the formal study of decision-making where several players must make choices that
potentially affect the interests of the other players. The object of study in game theory is the game,
which is a formal model of an interactive situation. It typically involves several players; a game with only
one player is usually called a decision problem. The formal definition lays out the players, their
preferences, their information, the strategic actions available to them, and how these influence the
outcome.
A game is a formal description of a strategic situation. A central assumption in many variants of game
theory is that the players are “rational”. A rational player is one who always chooses an action which
gives the outcome he most prefers, given what he expects his opponents to do. The goal of gametheoretic analysis in these branches, then, is to predict how the game will be played by rational players,
or, relatedly, to give advice on how best to play the game against opponents who are rational. It is
important to note that there is a dispute regarding the economists definition of rationality. In fact, Sen
in his article “Rational Fools” has said that people who always look at narrow self interest and attempt
to maximize it are Fools. “Rational Fools: A Critique of the Behavioral Foundations of Economic Theory,”
Amartya K. Sen, Philosophy and Public Affairs, Vol. 6, No. 4. (Summer, 1977), pp. 317-344. We use these
models of rationality to predict behavior in economic theory, but they don’t work too well in practice.
Section 1: Nodes and
Branches
Games in Tree form
The game tree, also called an
extensive form, is more detailed
than the strategic form of a game. It
is a complete description of how
the game is played over time. This
includes the order in which players
take actions, the information that
players have at the time they must
take those actions, and the times at which any uncertainty in the situation is resolved. This section treats
games of perfect information. In an extensive game with perfect information, every player is at any
point aware of the previous choices of all other players. Furthermore, only one player moves at a time,
so that there are no simultaneous moves.
Game trees are made up of nodes and branches, which are used to represent the sequence of moves
and the available actions, respectively.
Example 1:
Consider two players, Ahmad and Bilal, who are playing a sequential game. Ahmad moves first and has
the option of Up or Down. Bilal then observes his action. Regardless of what Ahmad chooses, he then
has the option of High or Low.
This is a game tree with perfect information. Every branching point, or node1, is associated with a player
who makes a move by choosing the next node. The connecting lines are labeled with the player’s
choices. The game starts at the initial node, the root of the tree, and ends at a terminal node, which
establishes the outcome and determines the players’ payoffs. In this subject, decisions are represented
by square nodes. Node a is the decision node (Any node where a decision is required by one of the
primary players) where Ahmad chooses between Up and Down. Since node a is the first node, it is also
known as the initial node. Nodes b and c are the decision nodes at which Bilal chooses between High
and Low. The triangle-shaped ending nodes on the right are the terminal nodes, which also have the
payoffs for each player associated with each outcome listed beside them.
Example 2: A game of Tic-Tac-Toe starts with a blank 3 x 3 board. The first player has NINE choices, since
he can choose to put an X in any one of the NINE blank squares. In practice there are only THREE choices
(corner, centre, and middle of an edge). Because the board is symmetric, all other choices are equivalent
to one of these three. A tree diagram for this first move looks like this:
1
A NODE is a point at which one of the players must make a decision. Each possible decision (or move)
corresponds to a branch going out of the node.
At each of these three new NODES of the game tree, the second player has options for the second
move. At the first node, corresponding to the CENTER move of the first player, there are EIGHT
branches. Each branch corresponds to putting an “O” in one of the eight available squares. Strategically,
there are only TWO moves: Corner and Edge, since all of the other moves are equivalent. Rotating the
board makes this clear. At the second node, corresponding to the CORNER move by first player, there
are eight branches, and FIVE different possible moves by the second player. The remaining three are
equivalent to one of the five. Exactly the same is true of the last node, corresponding to an EDGE move
by the first player. At this node there are Eight Branches and FIVE distinct and different moves.
Exercise: Game of Nim
NIM is a mathematical game of strategy in which two players take turns removing objects from a pile
containing N objects. On each turn a player must remove at least one object, and at most two objects.
The person who removes the last object wins the game.
DRAW the game tree for a game of NIM starting with N=5 objects. Can you tell which of the two players
will win if both play best strategies?
Example 3: Battle of Sexes:
Imagine a couple that agreed to meet this evening, but cannot decide if they will be
attending the shopping (S) or a football match (F). The husband would most of all like
to go to the football game. The wife would like to go to the shopping. Both would prefer to go to the
same place rather than different ones. To convert this real life situation into a game, we consider this as
a sequential game in which the Wife makes the first move by announcing her decision: S or F. At the
second move, the Husband announces his decision, also f or s (the lowercase letter distinguishes the
Husbands moves from the Wifes). The Game tree can be drawn as follows.
Exercise: Draw game tree, when the Husband moves first.
Section 2: Payoffs.
Payoffs are usually numerical, because economists assume everything can be translated into a single
utility. More recently, it has been discovered that this may not be possible and payoffs might occur on
different dimensions which cannot be combined in any one measure. However this has not yet had
impact on game theory, where we still use single numbers to describe payoffs.
Consider battle of sexes, the husband would most of all like to go to the football game. The wife would
like to go to the shopping. Both would prefer to go to the same place rather than different ones. We will
assign higher utility value/ payoff to husband when both of them meet at football game (2 vs3, here
(2,3) mean payoff of 2 for first player i.e., wife and 3 for second player means husband) and higher
payoff to wife if the meet at shopping. For separate places, both will get nothing (0,0).
Exercise: After Wife plays S, what is the payoff to Husband from playing s? What is the payoff to
Husband from playing f? Which of the two is better for the Husband?
Under assumptions of individualistic rational behavior (which is commonly used in game theory), each
player will make the choice which maximizes payoff to himself/herself. This assumption can often be
violated in real life situations. The object of behavioral game theory is to find out areas where game
theory assumptions are violated by actual human behavior, and then to find better theories which more
accurately explain behavior. For example, in this situation, if wife announces that she will go shopping,
then the optimal strategy for husband is to announce the he is going shopping. He gets a payoff of 2,
which is better than 0, which both would get if he goes to the football game. However, if Husband is
angry at wife for announcing shopping because she has been doing that for the past few days and
disregarded his preference for hunting in her selfish choice, he might choose to punish her by
announcing football, even though it brings lower payoffs to both. This would not match game theory
prediction, but could easily match real world behavior.
Exercise: REWRITE the payoffs in the game tree for Battle of Sexes under the following assumptions:
Husband gets utility of 3 from football and -1 from shopping. Wife gets utility of 4 from shopping and -2
from football. Husband gets utility of 6 from being together and Wife gets utility of 8 from being
together. Utilities are additive.
EXERCISE: In the game of NIM, there are five 100 Rupee notes on the table. Each player can take either
100 or 200 rupees on his turn. The player who takes the last note also gets a bonus prize or 500 Rupees.
Draw the game tree and the payoffs for this game.
Section 3: Solving Games by backwards induction.
Backward induction is the process of reasoning backwards in time, from the end of a problem or
situation, to determine a sequence of optimal actions. It proceeds by first considering the last time a
decision might be made and choosing what to do in any situation at that time. Using this information,
one can then determine what to do at the second-to-last time of decision. This process continues
backwards until one has determined the best action for every possible situation (i.e. for every possible
information set) at every point in time.
Backward induction assumes that players will move optimally at each node — that opponents can be
expected to act in their own best interests. Knowing this, a decision-maker working to solve a tree can
confidently eliminate actions that are suboptimal to his or her opponents.
Example 1: consider the earlier game
between Ahmad and Bilal.
At node b, playing High gives Bilal a payoff
of 0, while playing Low gives his a payoff
of 2. Therefore, Bilal would rationally
choose to play Low. We can ignore the
possibility of Bilal’s playing High at node b. Similarly, we can ignore the possibility that Bilal will play Low
at node c since his payoff for High is 1 and for Low is 0. In short, of the four strategies available to Bilal,
backward induction implies that his only rational strategy is to play Low at node b and High at node c.
This implies that Ahmad’s choices look as follows:
Notice that Ahmad's optimal strategy is now obvious — play Down. Down yields a payoff of 2 while Up
gives a payoff of only 1.
Example 2: Consider Battle of Sexes, starting backward induction we will first look at the terminal nodes
and the decisions of husband. He will as per assumption try to maximize his satisfaction by opting for
higher payoffs. Therefore, he will opt for football game at root node F by wife and chooses shopping at
node shopping by wife. Keeping in mind the choice of husband, Wife will also prefer to go for shopping,
resulting in the foreshown Nash equilibrium at (S,s), which is favorable to Wife. Similarly, if we do same
for game if husband moves first, we can notice that game will have NE at (F,f), which is favoring
Husband.
In this game, note that the first player has an advantage. Once the Wife commits to shopping, it
becomes optimal for the Husband to follow her strategy. Whoever moves first constrains the choices of
the other player. However, there are other games where the second player can have an advantage.
Example 3: Consider matching pennies game, in the matching pennies game, player A loses a point to B
if A and B play the same strategy and wins a point from B if they play different strategies. Since player B
can observe result of his coin toss, he is always better position to respond accordingly. In this game, it is
a disadvantage to be the first player. The second player always wins the game.
Example 4: NIM with THREE coins at start.
If there are three coins in the starting pile, then A will lose no matter how he plays. See that this is the
case by formally doing backwards recursion to solve the game. With Four coins, A can win. If he
removes one coin then B will become first player in a three coin game and lose. If A makes a mistake and
removes 2 coins then B can win by removing the last two.
EXERCISE: For any given number of coins N, determine whether the first player will win or lose, and also
explain what the optimal strategy is for both players.
It is important to note that this backward recursion method makes the assumption that every player is
rational and always pursues self interest. Game solutions can be different if this assumption cannot be
relied upon. Consider the following simple two player game:
Game theory predicts that player 1 will play HIGH. At second stage, P2 should also play high, since his
payoff of 1 is better than his payoff of 0, which occurs when he plays LO. HOWEVER, is this correct? Can
player I count on the rationality of player 2? What if player 2 makes a mistake, which will result in a loss
of 100? Shouldn’t player 1 play LOW just to be on the safe side? These issues are the subject of
behaviorial decision theory, which deals with how humans actually behave. These issues do not arise in
standard game theory, where it is assumed that all humans are “rational” in the peculiar sense that they
always maximize self-interest and do not consider other aspects of the game.
Section 4: Simultaneous moves
Consider battle of sexes simultaneous moves, here failure of coordination can cause problem.
Sequential moves are advantageous for BOTH players, since coordination failure will not occur. In game
with sequential moves, second player has already observed the action of first player so his information
set contains exactly one decision node. Note that there is an advantage to the first player in this game;
he or she can get to their desired outcome.
Here orange circle represents the information set of husband. Husband has to decide about his move
without knowing what move has been made by Wife; his information set contains two decision nodes.
Example: Matching Pennies Without Observation
Here player 1 moves first and puts down her rupee coin heads up or tails up. Player 2 does not observe
the outcome. He then moves putting down his rupee coin heads up or tails up. The game tree is as
shown below.
The game has the same structure as for the game with observation, however the two decision nodes
corresponding to player 2 have been encircled. A set of encircled decision nodes is called an information
set . When play reaches one of the decision nodes in an information set and it is the turn of that player
to move, the player does not know which of these nodes he is actually at. The reason for this is that the
player does not observe something about what has previously happened in the game.
Use information sets to draw the game tree for the following two simultaneous move games:
Exercises: Prisoner’s Dilemma Game payoffs.
The story behind the name “prisoner’s dilemma” is that of two prisoners held suspect of a serious crime.
There is no judicial evidence for this crime except if one of the prisoners testifies against the other. If
one of them testifies, he will be rewarded with immunity from prosecution, whereas the other will serve
a long prison sentence of five years. If both testify, their punishment will be less severe, say three years
each. However, if they both “cooperate” with each other by not testifying at all, they will only be
imprisoned briefly, say one year each, for some minor charge that can be held against them.
Draw game tree and write payoffs for prisoners’ dilemma.
Exercise: Blotto.
Colonel Blotto must defend two cities with one indivisible regiment of soldiers. His enemy, Colonel
Sotto, plans to attack one city with his indivisible regiment. City I has a value of 10 units, while City II has
a value of 5 units. If Colonel Sotto attacks a defended city, Sotto loses the battle and obtains nothing. If
Colonel Sotto attacks an undefended city, he obtains the value of the city, while Colonel Blotto loses the
value of the city. Neither Colonel knows what the other plans to do.
Draw game tree and write payoffs for Blotto Game.
5. Strategies:
A player's strategy in a game is a complete plan of action for whatever situation might arise; this fully
determines the player's behavior. A player's strategy will determine the action the player will take at any
stage of the game, for every possible history of play up to that stage. Students often confuse strategies
with choices corresponding to branches at a decision node of a game tree. A strategy is describes the
choice a player will make at every decision node on the entire game tree, and so is very different from a
choice at a particular node. We start with a simple example to illustrate the concept of a strategy.
Example 4: This game called “Ultimatum Game” was first introduced by Güth, Schmittberger and
Schwarze (1982). It is a sequential game instead of a static game. Suppose somebody gave you 300
rupees in 100 rupees note from (which means there are three 100 rupee notes) and asked you to divide
them with another person. Suppose one of you being the “proposer” and the other being the
“responder”. The proposer suggests a way to divide the notes (Rs, (300,0), (200,100), (100,200), (0,300),
four moves for first player and two for second player). If the responder accepts, then the money is
divided according to the proposer’s plan. If the responder rejects, then everyone gets zero. A game tree
for the ultimatum game is as follows:
The proposer has FOUR choices at the initial node: Offer 0,10,20, or 30. The proposers strategy is the
same as his choice because he has ONLY one decision. So all the decisions on all the nodes of the game
tree are reduced to one decision on one node.
The responder will actually have only one decision to make. He will be given an offer and then he will
have to decide whether to accept or reject. HOWEVER, a strategy for the responder specifies what he
will do at each node. There are four nodes, and two choices at each node, so responder has 2 x 2 x 2 x 2
= 16 strategies. A strategy can be coded as a collection of four letters: (RAAA). This strategy specifies the
response on each of thefour branches of the game tree. The first position is for the first node
corresponding to an offer of (30,0), which give ZERO to the responder. The R means we reject this offer.
The second position is for an offer of (20,10) which offers 10 to the responder. The A means we accept
this offer. Similarly the last two A’s mean that we also accept offers of 20 and 30 from the proposer. This
strategy is a rational strategy in the sense that all positive offers are accepted. However the concept of
strategy does not involve rationality. ALL possible plans of action are strategies. For example the
strategy (RRRR) specifies that we reject all offers is also a strategy. It is just a bad strategy from the point
of view of maximizing payoff to responder. However it might be strategy someone would use if he/she
was very angry with the proposer. Behavior need not be rational.
Example 2:
The game may describe the situation of, say, player I as a restaurant owner, who can provide food of
High quality (strategy H) or low quality (L), and a potential customer, player II, who may decide to eat
there (buy) or not (don’t buy). The customer prefers l to r only if the quality is good.
The restaurant owner , player I, makes the first move, choosing High or Low quality of service. Then the
customer, player II, is informed about that choice. Player II can then decide separately between buy and
don’t buy in each case. The resulting payoffs are the same as in the strategic-form game in Figure below:
The player I has one decision to made at first node, so his choices are his strategies (i.e., High and Low).
But the strategies of customer, player ii, are her decisions at each node. As she has two decision nodes,
her strategies will be represented by two possible actions. Since she has two choices at each node , she
will have 2*2=4 strategies in total i.e., (Buy,Buy), (Buy, Don’t buy), (don’t buy, buy), and (don’t buy,
don’t buy). Now (buy, don’t buy strategy) means “buy if offered high-quality service, don’t buy if offered
low quality service.”
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(Buy,Buy) is not an optimal strategy as player II buy even if services is of low quality.
(Buy, Don’t buy) is an optimal strategy as player II decides to buy if quality is high and don’t buy
if quality is low.
(don’t buy, buy) is not an optimal strategy as player II does not buy when quality is high and
buys if quality id low.
(don’t buy, don’t buy) is not an optimal strategy as player II does not buy even if quality of
service is high. She can actually increase her payoff buy buying when quality is high.
We can list down strategies of player I and II in a 2*4 table, where top left cell shows the payoff when
player I provides high quality service and player two has strategy (buy, buy).
Exercise: In the game of NIM with THREE coins to start, both players have essentially ONLY ONE NODE in
the game tree at which they have to make a choice. The first player at the first move must choose
between removing one or two coins. He does not get any further choices in the game. The second player
gets a choice ONLY if the first player removes ONE coin. In this case he can choose between one and two
coin removal. Of courses he will remove two since that wins the game, but he does have a choice. If the
first player removes two coins then there is no choice, because he must remove the one coin remaining
and win the game. SO in this game, strategy is same as choice because there is only one node of choice
for each player.
However if we start from four coins, then there are two choice nodes for each player. At each node, a
player has two actions. This means that there are four strategies for each player; call them A,B,C,D for
player 1 and (1,2,3,4) for player 2. Draw the game tree and list all four strategies for each of the two
players. ALSO draw a four by four diagram which shows what happens first player plays one of these
four strategies and the other player plays one of his four.