GAMES AGAINST NATURE
Topic #3
Games Against Nature
• In game theory, for reasons that will be explained later, the
alternatives (e.g., LEFT and RIGHT) that each player chooses
from are usually called strategies,
– but they may also be called choices, actions, etc.
• A Game Against Nature is
– a 1-player game, in which a single rational self-interested player must
choose a strategy, and
– the outcome and the player’s payoff depends on both his chosen
strategy and the “choice” made by a totally disinterested nature.
• A Game Against Nature part of what is generally called
decision theory (rather than game theory) because there is
only one player who
– makes a rational choice, and
– is interested in (gets a payoff, i.e., a gain or loss, from) the outcome.
A Decision Problem without Risk or Uncertainty
Choices for P1
=>
s1
s2
s3
s4
s5
•
Preferences/Payoffs/Utility
u1(x) = 4
u2(y) = 5
u3(z) = 3
u4(v) = 4
u5(w) = 3
the decision problem is conceptually trivial, i.e.,
P1 should choose the strategy that gives him the best outcome,
–
in this case s2.
However, such decision problems are often difficult in practice,
because the decision maker may not know how to predict or
evaluate the payoffs from the different outcomes,
–
•
x
y
z
v
w
=>
In this case, Nature makes no choice, so
–
–
•
Outcomes
but this problem is essentially one of information and research, not
strategic choice.
But decision/game theory focus on the problem of strategic
choice, and assumes that all players know their payoffs.
A Literal Game Against Nature
• A payoff matrix is set up as a crosstabulation of P1’s strategies
and Nature’s contingencies (or another player’s strategies).
– This is the simplest possible non-trivial matrix , with two rows and two
columns.
• Numbers in each cell of a payoff matrix indicate the payoff (or
at least rank the payoffs) to each player given by the resulting
outcome.
– In this case, there is only one number in each cell because there is only
one player, and
– nature is indifferent among outcome and has no payoffs.
The Maximax Principle
• We now identify several different decision principles (some
of which may be poor principles) that player P1 might use in
order to pick his strategy.
• The Maximax Principle (informally, “aim for the best”):
– choose the strategy that can give you the best outcome.
– In this case, maximax says “choose s2.”
• Note however that s2 can also give you your worst outcome,
– so maximax can mean “aim for the best --- but get the worst.”
The Maximin Principle
• Maximin Principle (informally, “avoid the worst”):
– For each strategy, determine the worst outcome (minimum payoff)
you could possibly get if you select that strategy;
• this is called the security level for each strategy.
– In this case ,
• the security level of s1 is 2, while
• the security level of s2 is 1.
– Then choose the strategy that gives you the highest security level,
• i.e., “the maximum of the minimums” or “maximin.”
– In this case, s1 is P1’s maximin strategy, and by using it P1 can
guarantee a payoff of at least 2, regardless of what Nature does.
Maximax
vs.
Maximin
• For reasons that will
be made evident
later, the term
‘minimax’ is used
more commonly
than ‘maximin.’
The Principle of Insufficient Reason
• The Principle of Insufficient Reason (informally, “maximize the
average payoff”):
– choose the strategy the gives you the highest average payoff (averaged
across contingencies).
– In this case, s1 and s2 have the same average value (2.5).
• This principle has two big problems:
– It assumes that the payoffs are cardinal [or interval], whereas
• the payoffs displayed in the matrix above are probably just ordinal.
– It also in effect assumes that all contingencies are equally probable.
Decision Making (cont.)
• In general, Nature’s choice of contingencies may be made
– by an objective and known (to the player) random mechanism (e.g.,
at a Las Vegas gaming table);
• This is called decision making under (objective) risk;
– or by some process concerning which the player has subjective
probability estimates (e.g., whether it will rain or shine);
• This is called decision making under (subjective) uncertainty.
• In either event, it is unlikely that all of nature’s contingencies are
equally probable.
Principle of Expected Utility Maximization
• Principle of Expected Utility (or Payoff) Maximization:
– The payoff matrix has been modified to show cardinal (or interval), rather
than merely ordinal, payoffs.
• Choose the strategy that gives you the highest expected payoffs, weighting
each contingency by its (objective or subjective) probability.
– This requires that
• payoffs are cardinal (or interval) in nature and
• the decision maker knows the probability over nature’s contingencies .
– Given p1 = p2 = .5, the expected payoff from s1 = 6 and from s2 is 5, so P1
should choose S1.
– If p1 = .25 and p2 = .75, the expected payoff from s1 is 7 and from s2 is 7.5,
so P1 should choose s2.
Expected Utility Maximization (cont.)
• If rain is certain, P1 definitely should take his umbrella
• If shine is certain, P1 definitely should not take his umbrella
• How probable does rain have to be so that P1 (if he is maximizing
expected payoffs) should choose to take his umbrella?
•
Put otherwise, what is the probability rain that leaves P1 indifferent between his two
strategies (because they have the same expected payoffs)?
• Let p (0 < p < 1)be the probability of rain, so the probability of shine is
(1-p).
•
•
Expected Payoff from S1 = Expected Payoff from S2
4p + 8(1-p) = 0p + 10(1-p)
6p = 2
p = 1/3 (and p = 2/3)
In which case the expected payoff from s1 = 4/3 + 16/3 = 20/3 = 7.333 ,
The expected payoff from s2 = 0/3 = 20/3 = 7.333
Eisenhower Before D-Day
• D-Day weather decision made by Supreme Allied Commander.
– D-Day June 5 was postponed due to a storm in the channel and another
storm was predicted to arrive shortly after the first.
– RAF Meteorologist James Martin Stagg predicted there would be a brief
window on June 6 with a break in weather.
• Eisenhower: “OK – let’s go.”
– Marginal weather predicted for June 6 may have been advantageous in
terms the (Zero-Sum) Game Against the Germans (as opposed to the Game
Against Nature).
– To invade only with favorable tides and full moon entailed disadvantages in
terms of the Game Again the Germans but evidently more than
compensating advantages in terms of the Game Against Nature.
• The Game Against the Germans was, in this respect, “strictly
determined” (as defined in Topic #5).
Eisenhower Before D-Day (cont.)
Eisenhower Before D-Day (cont.)
Cardinal Payoffs ≈ $$
•
•
Generally, you can think of cardinal payoffs as being essentially equivalent to
money, but not always.
Consider a game with two players plus Nature:
– P1 is you, deciding whether to buy home-owner’s (fire) insurance;
– P2 is an insurance company, deciding whether to sell you a policy;
– Nature decides whether your home will burn down in the next year.
•
If you and the insurance company have the same beliefs about the probability that
your house will burn down in the year, is there a premium price such that you will
buy and they will sell
– Not if payoffs = expected $$$.
– You probably are not risk-neutral but rather risk-averse.
– But the insurance company, insuring millions of widely dispersed house, can afford to be
risk neutral.
•
•
Ditto in a game between you, a Las Vegas gaming house, and Nature.
If you win the state lottery, are you indifferent between these two prizes?
– Prize 1: $1,000,000
– Prize 2: Another lottery ticket, which gives you a .5 chance of winning $2,000,000 and a
.5 chance of winning $0.
Decision Problems – Some Practice
Decision Problems – More Practice
• But can you see another and very persuasive decision
principle lurking in this example?
The Dominance Principle
• The Dominance Principle (D & N, Rule 2, p. 66):
– If strategy A gives you at least as good a payoff in every
contingency as strategy B and a better outcome in at least
one contingency,
• do not choose strategy B (as it is dominated by strategy A);
• choose an undominated strategy instead.
– In the payoff matrix above, s2 dominates both s1 and s3, so
P1 should not choose s1 or s3.
– Put otherwise, S2 is at least as good a “reply” as s1 (or s3) in
every contingency and is a better “reply” in at least one
contingency.
The Dominance Principle (cont.)
• Important note: the Dominance Principle says:
– if strategy A dominates strategy B, don’t use strategy B.
• The Dominance Principle does not say:
– if strategy A dominates strategy B, use strategy A, because
• there may be some strategy C that dominates A, or more likely
• there may be some strategy D that is undominated and which gives
a larger payoff than A in some contingency.
• In the matrix above, s2 and s4 are both undominated,
– that is, each is a best reply in some contingency.
The Dominance Principle (cont.)
• In the modified payoff matrix above, s2 now dominates s4
(as well as s1 and s3),
– so s2 is a dominant strategy.
• A dominant strategy if a best reply in every contingency.
• Corollary of Dominance Principle: If you have a dominant
strategy, use it.
• Notice that the Dominance Principle does not
– require cardinal payoffs in order for it to be applied, and
– does not depend on the probabilities of the contingencies.
The Dominance Principle (cont.)
• The Dominance Principle is consistent with every decision
principle discussed earlier.
• If s1 dominates s2,
–
–
–
–
s2 cannot be uniquely maximax;
S2 cannot be uniquely minimax
S2 cannot have a higher average payoff
S2 cannot have a higher expected payoff.
• But typically players may have many (even only) undominated
strategies and
– only rarely do players have dominant strategies.
• Therefore, while the Dominance Principle is very good advice
where it applies, it very often does not apply.
The Dominance Principle (cont.)
• The Dominance Principle appears to be a good principle to
follow, not only in Games Against Nature, but also in Games
Against Other Players.
– This is not true of the other Decision Principles.
– But notice that RIGHT dominates LEFT in the Social Dilemma Game
(and compare with the Prisoner’s Dilemma Game in Topic #4).
• Two examples of dominant strategies (one trivial, one tragic):
– Following another car into an almost full parking garage.
– Passengers on United #93.
• A threat to use a dominant strategy is highly credible (as will be
discussed in Section II of the course).