5.4
Simultaneous Games
[5.2 Using Game Theory] [5.3 Classic Game Models]
[5.5 Sequential Games] [5.6 Oligopoly]
Simultaneous (or static)
games are games in which all players try to make decisions based on what each
thinks the others will do and in which no player has any information about the
actions of the other players. Each player has to base his or her actions on
what he or she thinks (or anticipates) the other player will do.
Note that the payoff numbers provided in the various examples presented in this section are arbitrary and only intended to illustrate the concept, not any actual result.
The concept of equilibrium
is central to game theory. As stated earlier, equilibrium occurs when each
player is using the strategy that will provide them with the optimal payoff
given the strategies of all other players. Understanding and employing the idea
of equilibrium in games is what allows you to make the right strategic choice
for any given situation.
An equilibrium strategy
provides the highest payoffs to each player given the conditions of that
particular game. However, in game theory, terms such as "highest" and
"optimal" do not always mean "good". In fact, as in the
classic Prisoner's Dilemma game, where equilibrium occurs when both players
confess, "optimal" can even mean "not good". Equilibrium
strategy simply points you to the best move you can make given the anticipated
or actual moves of your opponents.
A common way of
representing games, especially simultaneous games, is the normal form representation, which
uses a table structure called a payoff matrix to represent the available strategies (or actions)
and the payoffs.
For illustrative purposes, suppose that we have two players — Albert and Betty
— who are playing a simultaneous game.
Albert has the choice of
two strategies — moving up or down — while Betty has the choice of left or
right. If Albert chooses up and Betty chooses left, Albert receives $20 and
Betty receives $20. If Albert and Betty choose up and right, respectively, they
receive $20 and $10, respectively. For down and left, the payoffs are $9 and
$20. For down and right, the payoffs are $10 and $40. The payoff matrix for
this game would look as follows:

Betty 

Left 
Right 

Albert 
Up 
$20, $20 
$20, $10 
Down 
$9, $20 
$10, $40 
This is a conventional method of representing the payoffs with the payoff of the player on the left of the matrix (Albert) listed first and the payoff of the player on the top of the matrix (Betty) listed second.
Suppose that we were to
allow an additional choice for Albert and Betty so that they could now choose
to play middle. The new payoff matrix would look like this:

Betty


Left 
Middle 
Right 

Albert 
Up 
$20, $20 
$30, $25 
$20, $10 
Middle 
$11, $25 
$9, $20 
$30, $10 

Down 
$9, $20 
$11, $20 
$10, $40 
Thus, if Albert and Betty
both play middle, their payoffs are $9 and $20, respectively.
Once a game has been
modelled or represented, as in the above tables, the next thing you want to do
is construct a strategy for maximising your payoff, given the variables of the
particular game you are playing. Maximising your payoff is different to beating
your opponent. In some settings, particular combinations of strategies may lead
to higher payoffs for all players than other combinations so doing better than
other players may not be consistent with maximising your payoff.
NOTE: There are actually two kinds of dominant
strategies: strict and weak. The distinction between the two can be important
in some areas but in this subject, when the term "dominant strategy"
is used, the reference is to strictly dominant strategies.
In simultaneous games, the
clearest move a player can make is to follow what game theory refers to as his
or her dominant strategy, if one exists. In game theory, rational
players can always be counted on to follow their dominant strategy.
A dominant strategy is a
strategy that provides a player with the highest possible payoff for any
strategy of the other players. Note that a dominant strategy does not
necessarily provide a player with a higher payoff than the opponents. Being
dominant simply means that strategy provides a higher payoff than any other
strategy that player could use given the anticipated actions of the opponents.
To clarify exactly what
this means, consider the following advertising game between competing sellers
of bottled water — Kind Lake Bottlers and Red Dolphin Sparkling Water. Suppose
that each has to decide between spending a moderate or large amount on
advertising. This is a simultaneous game with two players, each of whom has two
options. A normal form representation of the game is shown below:

Red Dolphin 

Moderate 
Large 

Kind Lake 
Moderate 
$80, $100 
$40, $80 
Large 
$110, $65 
$55, $40 
Consider the game from the
perspective of Kind Lake first. If Red Dolphin is expected to choose a moderate
amount of advertising expenditures, Kind Lake's best response is to spend a
large amount and earn a profit of $110 (rather than $80). If Red Dolphin is
expected to spend a large amount, then Kind Lake's best response is still to
spend a large amount and earn $55 (rather than $40). So Kind Lake should spend
a large amount regardless of what Red Dolphin does. This means that for Kind
Lake, spending a large amount is a dominant strategy — a strategy that provides
the highest payoffs no matter what Kind Lake anticipates Red Dolphin will do.
Note: All dominant strategies are optimal,
but not all optimal strategies are dominant. Why? By definition, a dominant
strategy is always a player's best move; therefore, it is optimal. However,
when a player does not have a clearly dominant strategy, he or she must still
choose the strategy that gives the best ("optimal") chance of
maximising payoffs. (See the Successive Elimination of Dominated Strategies
animation.)
Now consider the game from
the perspective of Red Dolphin. If Red Dolphin's management considers it likely
that Kind Lake will spend a moderate amount, Red Dolphin's best response is to
also spend a moderate amount and earn $100 (rather than $80). If it is
anticipated that Kind Lake will spend a large amount, Red Dolphin's optimal
move will again be to spend a moderate amount and earn $65 (rather than $40).
So, for Red Dolphin, moderate spending is a dominant strategy. To maximise its
payoff, Red Dolphin should spend a moderate amount no matter what it
anticipates Kind Lake will do.
In this case, Kind Lake
choosing a large amount and Red Dolphin choosing a moderate amount is the
equilibrium for the game above, as neither grocery would have any desire to
switch its choice given the expected choice of the other.
When a strategy is
dominated, it simply means there is at least one other strategy available that
offers a higher payoff for each strategy of the other player. A player seeking
to maximise payoff should never play a dominated strategy since, by definition,
there is always a better or more beneficial strategy available to them in that
game. As an example, return to the game between Kind Lake Bottlers and Red
Dolphin Sparkling Water.

Red Dolphin 

Moderate 
Large 

Kind Lake 
Moderate 
$80, $100 
$40, $80 
Large 
$110, $65 
$55, $40 
From Kind Lake's
perspective, large spending has already been established as the dominant
strategy because it offers higher payoffs ($110 or $55) than moderate spending
($80 or $40) no matter what Red Dolphin is expected to do. So for Kind Lake,
moderate spending is a dominated strategy — under no circumstances will Kind
Lake be better off playing moderate instead of large in this game.
Now looking at the table from Red Dolphin's perspective, you see that large is the dominated strategy for similar reasons: under no circumstance will Red Dolphin be better off playing large when it can play moderate — its dominant strategy.
Equilibrium is arrived at
differently in sequential and simultaneous games. In some sequential games,
players can reach equilibrium by simply responding rationally to an opponent's
previous move and anticipated subsequent actions. In simultaneous games, the
equilibrium is known as Nash equilibrium and is often harder to attain.
Nash equilibrium describes
a state in a simultaneous game at which each player has chosen a strategy that
optimises his or her payoffs, given the strategies of all other players. What
this means is that in a state of Nash equilibrium, no player has an incentive
to choose any other strategy because, of all those available, they have chosen
the one that provides the optimal payoff — given the anticipated moves of other
players. In other words, if you as a player find a game's Nash equilibrium, you
find your best move.
Consider the earlier
example of a game between Albert and Betty:

Betty 

Left 
Right 

Albert 
Up 
$20, $20 
$20, $10 
Down 
$9, $20 
$10, $40 
By studying the payoff
matrix, Betty can anticipate that Albert will always go up because, for Albert,
up dominates down; that is, regardless of which strategy Betty chooses,
Albert's optimal strategy is to play up. So, playing up is a dominant strategy
(and playing down is a dominated strategy) for Albert. Believing then that
Albert will choose up, which move should Betty make: left or right?
Because she anticipates
Albert will go up, the Nash equilibrium strategy for Betty is to move left
where she will receive a payoff of $20 rather than $10. Here, neither player
would have any reason to want to switch his or her move based on the move of
the other.

Betty 

Left 
Right 

Albert 
Up 
$20, $20 
$20, $10 
Down 
$9, $20 
$10, $40 
NOTE: American mathematician John Nash is credited
with taking the idea of game theory and transforming it into a powerful and
useful analytic tool. While at Princeton in 1950, he wrote a thesis that first
introduced the concept of Nash equilibrium. This groundbreaking work was so
influential that Nash, along with two other game theorists, John Harsanyi and Reinhard Selten, was awarded the 1994 Nobel prize for economics.
Test your ability to find
Nash equilibria in the following exercise. Click here to launch the exercise.
Often in simultaneous
games, players have more than one Nash equilibrium. In these cases, it can be
much harder to narrow down all players' options to just one clear strategy.
Suppose, for instance,
Albert and Betty are talking on the phone but suddenly get disconnected. If
Betty tries to call Albert back, then Albert should stay off the line rather
than try to call Betty. However, if Albert decides to wait for Betty to call —
but Betty also decides to wait, thinking that Albert will call — the call will
never be completed. So the two Nash equilibrium outcomes for this game are
Betty calls/Albert waits or Betty waits/Albert calls, as shown in the following
payoff matrix:

Betty 

Call 
Wait 

Albert 
Call 
0, 0 
10, 10 
Wait 
10, 10 
0, 0 
Notice that with multiple
Nash equilibria neither player has a dominant
strategy. This illustrates the limits of the Nash concept: once multiple equilibria have been identified, the concept of Nash
equilibrium cannot be used to determine which equilibrium will actually occur —
or, in fact, if any will occur.
In the real world,
communication, even with an opponent, can often be the solution to this
problem. However, in situations where communication is not possible, the
determination of what outcome (Nash or otherwise) will occur is entirely
dependent on the overall conditions of the game.
For instance, in the game above, if Albert initiated the first call, it might be logical for him to initiate the second. However, maybe Betty knows Albert's phone number, but he does not know hers, in which case she has to be the one to call. Without clear communication here, the outcome is in question.
Often a player does not
have a clearly dominant strategy — making that game's equilibrium harder to
find. In such cases, the next best method a player can use is what game theory
refers to as "successive elimination of dominated strategies". This
rather lengthy and awkward phrase describes a process by which a player
eliminates all strategies that are dominated by some other available strategy
that player could adopt.
To illustrate this concept,
consider once again the advertising game between Kind Lake Bottlers and Red
Dolphin Sparkling Flowers. This time, there will be a new series of payoffs.

Red Dolphin 

Moderate 
Large 

Kind Lake 
Moderate 
$80, $60 
$10, $100 
Large 
$50, $10 
$40, $30 
Consider Red Dolphin first.
If Red Dolphin anticipates that Kind Lake will choose moderate expenditures,
the best action for Red Dolphin is to choose large expenditures and earn $100
(rather than $60). If Kind Lake is expected to choose large expenditures, the
best action for Red Dolphin is still to choose large expenditures and earn $30
(rather than $10). Therefore, a large advertising budget is a dominant strategy
for Red Dolphin, and moderate is a dominated strategy for Red Dolphin:

Red Dolphin 

Moderate 
Large 

Kind Lake 
Moderate 
$80, 
$10, $100 
Large 
$50, 
$40, $30 
Now consider Kind Lake. If Kind Lake anticipates Red
Dolphin will choose a moderate budget, the best action for Kind Lake is to
choose a moderate budget and earn $80 (rather than $50). However, if it is
believed that Red Dolphin will choose large expenditures, the best action for
Kind Lake is now to choose large and earn $40 (rather than $10). Therefore, the
best response for Kind Lake changes and is contingent on the choice made by Red
Dolphin. Kind Lake does not, in this case, have a dominant strategy since, as
noted above, a dominant strategy has the highest payoff regardless of the moves
of a competitor.

Red Dolphin 

Moderate 
Large 

Kind Lake 
Moderate 
$80, $60 

Large 

$40, $30 
So for Kind Lake, constructing its best strategy, given the payoffs in the above representation, requires it to use successive elimination of dominated strategies.
Since Kind Lake's
management can see that moderate expenditures is a dominated strategy for Red
Dolphin, they can assume that Red Dolphin will not choose moderate no matter
what it anticipates Kind Lake will do. Given this, Kind Lake can remove from
consideration the option that Red Dolphin will choose moderate. Each time a
player eliminates a strategy, the game is changed and prior analysis may no
longer apply. Here is the revised matrix with moderate eliminated for Red
Dolphin:

Red Dolphin 

Large 

Kind Lake 
Moderate 
$10, $100 
Large 
$40, $30 
Knowing now that Red
Dolphin's only rational option is to choose large, Kind Lake's best response is
to also choose large because its payoff in that case is $40, whereas if it
chose moderate its payoff would be only $10. Kind Lake anticipates that Red
Dolphin will not play a dominated strategy and, therefore, chooses its optimal
action, given that it anticipates Red Dolphin will choose large.

Red Dolphin 

Large 

Kind Lake 
Moderate 

Large 
$40, $30 
The following
animations show how players use successive elimination of dominated strategies
in a more complex game. Click here to view how
Albert eliminates a dominated strategy.
To continue the elimination of dominated strategies and
find the Nash equilibrium of the game, view the animation here.
Work through a game theoretic case by clicking on the link here.
The Practicalities of Cooperation
When players communicate, natural barriers preventing those players from
achieving higher payoffs can sometimes be removed. Many times, communication
simply brings relevant information to light, allowing all players to see the
game from the same perspective. Other times, however, communication is intended
to bring about cooperation — to get all players to act in a mutually
beneficial manner.
But is cooperation ever really practical in a business setting? Will
explicit or implicit cooperation between players ever result in the desired
outcome? Game theory demonstrates that the answer to these questions is usually
— but not always — no.
Suppose that on the hottest day of the year an outdoor festival is held.
There are two lemonade vendors at the festival. Knowing that their product will
be in high demand, the two vendors agree the night before to make less lemonade than what could be sold.
If they restrict their output (reduce the supply), they know prices will go up.
Then, if they are able produce less but charge more, both should enjoy
considerable profits — if they stick to the agreement. But how likely is it
they will stick to the agreement? Will they actually cooperatively restrict
their output in order to raise prices, or will they cheat and overproduce?
Consider the following payoff matrix:

Vendor B 

Cooperate

Cheat 

Vendor A 
Cooperate 
$20, $20 
$8, $28 
Cheat 
$28, $8 
$15, $15 
As you can see, each vendor has a dominant strategy to overproduce. (You
may recognise this as a Prisoner's Dilemma game.) Do the payoffs in the matrix
above make sense? Consider the case where both vendors cooperate. Here, the
vendors are effectively acting as a monopoly and splitting the profits. If both
overproduce, the vendors again split the profits. However, this time, they
would be in a competitive market and so the overall profits are lower. If one
of the vendors deviates, then that vendor will increase profits at the expense
of the other vendor.
Clearly, without a binding aspect to the agreement, it will not work in
most cases — the incentive to deviate, without enforceable consequence, is
usually too great for any other outcome
to occur.
The Organisation of
Petroleum Exporting Countries (OPEC) is one of the world's most successful
cartels. Since 1960, member countries have met at least twice a year to agree
on oil prices and set production quotas. By collectively restricting the amount
of oil member countries allow onto the market, OPEC is able to keep oil prices
(and profits) artificially high.
Without the OPEC agreement,
the individual countries would be forced to compete with one another directly
in the market, which would cause the prices to come down and the cartel's
inflated profits to drop. Knowing this, one could assume that the individual
members would be willing to do whatever it takes to keep their cartel healthy.
However, in the early
1980s, member countries in need of cash began to deviate from their quotas with
regularity. This deviation placed strong downward pressure on the price of oil,
which, in turn, forced the nondeviating countries, such as Saudi Arabia, to
actually reduce their own output (and therefore, profits) in an attempt to prop
prices back up. The deviating countries were essentially stealing from their
partners.
Adding to the problem of
overproduction was the willingness of some member countries, such as heavily
indebted Nigeria, to openly undercut OPEC's official, agreedupon prices
whenever the move suited them. These kinds of unauthorised price cuts
eventually led to a period of direct competition between cartel members that
saw prices fall from $30 a barrel in November of 1985 to a low of only $6 a barrel
in July of 1986.
OPEC asked the deviators to
stop time and time again but, short of declaring war, there was not much the
organisation could do to stop its less considerate members, all of whom were
sovereign states legally beholden to no one. Had OPEC punished these countries
by expelling them from the cartel, it would have lost any hope of controlling
them and the cartel's ability to bargain with the rest of the world would have
been weakened.
Here you can see that even
in the world's most "successful" cartel, cooperation between players
does not always work. This is because, while it is in the collective interest
of all members of the cartel to abide by production quotas and price
agreements, it is not in their individual interest to do so. Because all cartel
members have an incentive to overproduce or slash prices — absent enforceable
threats to punish a cartel member for such behaviour — they will do so whenever
the need arises.
OPEC will probably not fall
apart any time soon — its members enjoy too many benefits to allow that — but,
given human nature, it will also never be the truly cooperative organisation
its founders envisioned.
NOTE: Certain business activities are regulated under
international and US federal and state laws, including cooperation among
competitors. If you are considering any cooperative activity with one or more
competitors, you should consult with appropriate legal experts regarding the
legality of such actions.
In this
topic, you have learnt how to
·
formulate
simultaneous move games
·
identify
dominant strategies
·
find
the Nash equilibrium or equilibria of games
·
solve
a game by the successive elimination of dominated strategies
·
consider
the situations in which cooperation is both desirable and can be achieved
Now go on
to topic 5.5, “Sequential Games”.