Economics Exam notes
Chapter 4
Games are said to have simultaneous moves if players must move without the
knowledge of what their rivals have chosen to do. A game is also simultaneous when
players choose their actions in isolation, even if the choices are made at different
hours no player will have information about what the other have done or will do. Thus
simultaneous-move games have imperfect information.
Strategy is a complete plan of action. In a simultaneous game, each player has
at most one opportunity to act, whereas in a sequential they have multiple
opportunities. Therefore there is no distinction between a strategy and action in
simultaneous-move games.
Mixed strategies: A strategy can be a probabilistic choice from the basic
actions initially specified (for ex: in sports where players deliberately randomize their
choice of action to keep opponent guessing). Pure strategies are the basic initially
specified actions.
Simultaneous-move games with discrete strategies are most often depicted
with the use of a game table (aka game matrix or payoff table). The table is called
the normal form or the strategic form of the game.
Zero-sum or constant-sum game: are games where the interests of the two
sides are exactly the opposite of each other, therefore for each combination of the
player’s choices, the payoffs of one can be obtained by reversing the sign of the
payoffs to the other.
Each player wants to pick an action that yields her the highest payoff. Nash
equilibrium is a list of strategies, one for each player, such that no player can get a
better payoff by switching to some other strategy that is available to her while all the
other players stick to the strategies specified for them in the list.
The definition of Nash equilibrium does not require equilibrium choices to be
strictly better than other available choices. Figure 4.3: given column’s choice of
middle, row could not do any better than she does when choosing low so low, middle
qualifies for a Nash equilibrium.
More important, it does not have to be jointly for the players. In figure 4.1
(bottom, right) gives payoffs 9,7 which are better for both players then 5,4 of the nash
equilibrium. However playing independently, they cannot sustain (bottom, right).
Given that column plays right, row would want to deviate from bottom to lo and get
12 instead of 9. Getting the jointly better payoff of 9,7 would require cooperative
action that made such cheating possible.
Nash equilibrium can be found using cell-by-cell inspection, or enumeration.
People play simultaneous-move games and do make choices and do replaces the
actual knowledge or observation of others’ actions players can make blind guess. A
more systematic way to figure what others are doing is experience and observation for
instance- if players play this game or similar ones with similar players all the time
they may develop a pretty good idea of what others do. Another method is the logical
process of thinking through others’ thinking- as in out yourself in their position.
Even when all the rules of a game are known without any uncertainty external
to the game each player may be uncertain about what actions the others are taking at
the same time. Players choose in the face of this strategic uncertainty by forming
some subjective views or estimates about the others actions- that is what the notion of
belief captures. If a player does not know the actual choices of the others but has
beliefs about them, in Nash equilibrium those beliefs would have to be correct. Thus
Nash equilibrium is a set of strategies one for each player such that, 1) each player has
correct beliefs about the strategies of the others, 2) the strategy of each is the best for
herself given her beliefs about the strategies of the others.
The way of thinking about Nash equilibrium has two advantages, 1) concept
of best response is no longer logically flawed- each player is choosing her best
response, not to the as yet unobserved actions of the others but only to her own
already formed beliefs about their actions. 2) Where we allow mixed strategies the
randomness in one player’s strategy may be better interpreted as uncertainty in the
others players’ beliefs about this players’ action.
Some games have a special property that one strategy is uniformly better than or
worse than another. The well-known game of the prisoners’ dilemma illustrates this.
Each player has to choose between confessing and not confessing- they both know
that no confession leaves them each with a 3-year jail sentence for involvement of the
kidnapping. Also if one of them confesses, he or she will get a short sentence of 1
year for cooperating with the police, while the other will go to jail for a minimum of
25 years. If both confess, they figure that they can negotiate for jail terms of 10 years
each.
 Suppose he believes she will confess: then his best choice will be to confess
(payoff of 10 instead of 25)
 Suppose he believes she will deny: then his best choice is also to confess (payoff
of only 1 instead of 3)
Thus in this game confess is a dominant strategy or the strategy deny is a
dominated strategy because regardless of what the husband thinks the wife will do
the strategy confess will be better for him.
Prisoner’s dilemma game has three essential features:
1. Each player has two strategies- to cooperate or defect
2. Each player has a dominant strategy
3. The dominance solution equilibrium is worse for both players than the nonequilibrium situation in which each plays the dominated strategy.
Both players follow conventional wisdom in choosing their dominant strategy but the
resulting equilibrium outcome yields them payoffs that are lower than they could have
achieved if they had each chosen their dominated strategy- the problem is how could
they guarantee that someone will not cheat.
In some games only one player has a dominant strategy. In this case the other
rational player can assume that she will use this dominant strategy and so choose her
equilibrium action accordingly.
In larger games, some of a player’s strategies may be dominated even though
no single strategy dominates all of the other. In this type of game they may be able to
reach equilibrium by removing dominated strategies from consideration as possible
choices. This reduces the size of the game and the “new” game may have another
dominated strategy, which can be reduced until no further ones are available. Or the
new game may have a dominant strategy for one of the players. This is known as
successive or iterated elimination. If this process ends in unique outcome, then the
game is said to be dominance solvable.
Column
Left
Middle
Right
Top
3,1
2,3
10,2
Row
High
4,5
3,0
6,4
Low
2,2
5,4
12,3
Bottom
5,6
4,5
9,7
The only dominated strategy for row is high which is dominated by bottom; column’s
left is dominated by right. We must note that we could not have eliminated column’s
left unless we eliminated row’s high because against row’s high, column would get 5
from left but only 5 from right. Then within the remaining set of strategies (top low
and bottom for row, idle and right for column), row’s top and bottom are both
dominated by low. When row is left with only low, column chooses his best responsemiddle. The game is hence dominance solvable and the outcome is low middle- being
the Nash equilibrium.
Weakly dominated and strictly dominated strategy:
Top
3,1
2,3
10,2
Low
2,2
5,4
12,3
Bottom
5,6
5,5
9,7
Low dominated top, but the dominance of low over bottom is less clear. The 2
strategies give row equal payoffs against column’s middle, although low does row a
higher payoff than bottom when plays against column’s right. So from row’s
perspective low weakly dominates bottom and low strictly dominates top.
Column
Row
Left
0,0
1,1
Up
Down
Right
1,1
1,1
For row up is weakly dominated by down; if column plays left then row gets the same
payoff from her 2 strategies. For column right weakly dominates left, dominance
solvability then tells us that (down, right) is a Nash Equilibrium. That is true but down
left and up right are also Nash equilibrium. When row is playing down column cannot
improve by switching to right and when column is playing left rows best response is
down. Same thing applies up right. Therefore if you use weak dominance to eliminate
some strategies it is a good idea to make cell by cell check to make sure you haven’t
missed any other equilibria.
Many simultaneous-move games have no dominant or dominated strategies.
Others may have one or several dominated strategies but iterated elimination of
dominated strategies will not yield a unique outcome. In such cases best response
analysis is the tool- so we find each player’s best response strategy depending on the
other player’s available strategies. Best-response analysis is a comprehensive way of
locating all possible Nash equilibria of a game. In the below figure: if column chooses
left row’s best response would be bottom, for middle best respone is low… all
highlighted in green. For column, the best responses are circled in yellow; if row
chooses top- column will choose middle… etc. there will be some games for which
best-response analysis does not find a Nash equilibrium just as dominance solvability
sometimes fails. But when best-response of a discrete strategy game does not find a
Nash equilibrium then the game has no equilibrium in pure strategies.
Row
Top
High
Low
Bottom
Left
3,1
4,5
2,2
5,6
Column
Middle
2,3
3,0
5,4
4,5
Right
10,2
6,4
12,3
9,7
For zero-sum games where strict conflict exists, the minimax method is used
and only works for this game; it relies on a thought process that accounts for the fact
that outcomes that are good for one player are by definition bad for the other. In this
method each player asks herself “ is this the best choice for me, even if the other
player found out that I was playing it?” Then she must consider her opponents best
response to her chosen strategy. But in zero sum games, that best response is the
worst one for her; i.e. in 0-sum games each player believes that her opponent will
choose an action that yields her the worst possible consequences of each of her own
actions. Then acting on those beliefs she should choose the action that leads to the
least bad outcome.
Suppose row wants the outcome to be a cell with as high number as possible, column
would therefore want the opposite. Row figures that for each of her rows column will
choose the column with the lowest number in that row. Therefore row should choose
the row that gives her the highest amongst these lowest numbers.- the maximum
among the minimum—the maximin. Similarly column knows that for each of her
column, row will choose the row with the highest number in that column so column
should choose the column with the smallest number among the largest ones- the
minimum among the maxima—the minimax.
Defense
Offence
Run
Short pass
Medium pass
Long pass
Run
Pass
Blitz
2
6
6
10
5
5.6
4.5
3
13
10.5
1
-2
Begin by finding the lowest number in each row- the offence’s worst payoff from
each strategy- and the highest number in each column- the defense’s worst payoff
from each strategy. The offense’s worst payoffs are highlighted in yellow, and
defense’s are in green. The next step is to find the best of each player’s worst possible
outcomes. The largest of the row minima is 5.6, so that is the offence’s maximin. The
smallest of the column maxima is 5.6 so that’s the defense’s minimax. Thus the
offence’s maximin strategy is its best response to the defense’s minimax and vice
versa. In cases where the minimax method fails to find equilibrium in zero-sum
games we conclude that the game has no Nash equilibrium in pure strategies.
Three player games: suppose Emily is contemplating the possible outcomes
of the street garden game, there will be six possibilities- she can choose to either
contribute or not when both Nina and Talia contribute, or when neither contribute or
when just one of them do. For her, it’s best to take advantage of her neighbors that
contribute while she doesn’t (payoff of 6). She enjoys a bigger garden if she also
contributes but at the cost of her own contribution (payoff of 5). At the other end of
the spectrum are the outcomes that arise when neither Nina nor Talia contribute; here
Emily would again prefer not to contribute (payoff of 2 instead of 1 if she contributes
while they don’t). In between these cases are the situations in which either Nina or
Talia contributes but not both. In this case Emily prefers not to contribute (4) rather
then to contribute (3) and gain a larger garden. Because Nina and Talia have the same
views on the costs and benefits of contribution and garden size, each of them order
the different outcomes in the same way- the worst being the one in which each
contributes while the other 2 don’t. To find the Nash equilibrium we need a
game table:
Talia chooses
Contribute
Don’t contribute
Nina
Emily
Contribute
Don’t
Contribute
Don’t
5,5,5
6,3,3
3,6,3
4,4,1
Nina
Emily
Contribute
Don’t
Contribute
Don’t
3, 3, 6
4,1,4
1,4,4
2,2,2,
The first test should be to determine whether there are dominant strategies for any of
the players. For Emily we compare the two rows of both pages of the table and when
Talia contributes Emily has a dominant strategy not to, and when Talia doesn’t
Emily’s dominant strategy is also no to. Thus it is best for Emily not to contribute
regardless of what the other two players choose to do. The same applies to Nina. For
Talia, we must compare cells across pages of the table- the top left in first page with
that in second and so on. This indicates that Talia also has a dominant strategy not to
contribute. This game is another example of a Prisoner’s dilemma- there is a unique
Nash equilibrium where they all receive payoff of 2 but another outcome exists in the
game where all three yield higher payoffs of 5. Even though it is better for them to
pitch in and build the garden no one has the individual incentive to do so.
The NASH equilibrium of the game can also be found using cell-by-cell inspection.
For example consider cell (3,3,6) when Emily considers changing her strategy she can
only move to the next row but within the same column and page; Nina can change her
strategy to another column but in the same row and page; finally Talia can change
only the page position. While Emily and Nina improve their strategy from 3 to 4,
Talia become worse off from 6 to 5, but since at least one can become better off by
changing their strategy the cell that we examined is not Nash equilibrium.
We can also use best response strategy. Emily’s best response is highlighted yellowNina green- and Talia pink. The cell at bottom right has all three best responses and
therefore it gives us the Nash equilibrium.
Some games have no pure strategy Nash equilibrium and an example is that of
a single point in a tennis match. Given that down the line passing is stronger than a
cross-court shot and Evert is more likely to win the point when Navratilova moves to
cover the wrong side of the court a reasonable set of payoffs can be worked out. Evert
is successful with DL 80% of the time Navratilova covers CC, and 50% of the time if
N covers DL. Similarly, E is successful with her CC 90% if N covers DL and 20% of
the time when N covers CC. clearly this is a zero sum game because the fraction of
time that N wins this tennis point is just the difference between 100% and the fraction
of time that E wins.
Evert
DL
CC
Navratilova
DL
50
90
CC
80
20
From the above table, the rules for solving simultaneous-move games tells us to
look first for dominant or dominated strategies and then try to minimax or use
cell by cell inspection to find the Nash equilibrium. No dominant strategy exist
here- going on cell by cell we start with the choice of DL for both players. From
that outcome, E can improve her success from 50 to 90 by choosing CC. but then
N can hold E down to 20 by choosing CC. after this E can raise her success again
to 80 by making her shot DL and N in turn can do better with DL. In every cell
one player always wants to change her play and we cycle through the table
endlessly. What is important games with no Nash, is not what players should do
but what they should not do. Thus in this case players should not act
systematically, because their opponent can figure out the pattern of their actions
and decrease the chances of their success. In this case E for instance should
randomize her actions and this is known as mixing strategies.