Zero-sum Games
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The Essentials of a Game
Extensive Game
Matrix Game
Dominant Strategies
Prudent Strategies
Solving the Zero-sum Game
The Minimax Theorem
The Essentials of a Game
1. Players:
We require at least 2 players (Players choose
actions and receive payoffs.)
2. Actions:
Player i chooses from a finite set of actions, S =
{s1,s2,…..,sn}. Player j chooses from a finite set of
actions T = {t1,t2,……,tm}.
3. Payoffs:
We define Pi(s,t) as the payoff to player i, if Player i
chooses s and player j chooses t. We require that
Pi(s,t) + Pj(s,t) = 0 for all combinations of s and t.
ZERO-SUM
4. Information: What players know (believe) when choosing
actions.
The Essentials of a Game
4. Information: What players know (believe) when choosing
actions.
Perfect Information:
Players know
• their own payoffs
Common
Knowledge
• other player(s) payoffs
• the history of the game, including other(s) current action*
*Actions are sequential (e.g., chess, tic-tac-toe).
Extensive Game
Player 1 chooses
Player 2
Player 1
a = {1, 2 or 3}
b = {1 or 2}
c = {1, 2 or 3}
Payoffs = a2 + b2 + c2
-(a2 + b2 + c2)
Player1’s
decision nodes
1
“Square the Diagonal”
(Rapoport: 48-9)
if /4 leaves remainder of 0 or 1.
if /4 leaves remainder of 2 or 3.
1
2
3
Player 2’s
decision nodes
2
3
1
2
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22
GAME 1.
Extensive Game
How should the game be played?
Solution:
Strategy:
a set of “advisable” strategies, one for each player.
a complete plan of action for every possible decision
node of the game, including nodes that could only
be reached by a mistake at an earlier node.
Start
at the advisable
final
Player1‘s
decision nodes (in red)
Strategy in red
1
1
2
3
Backwards-induction
2
3
1
2
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22
GAME 1.
Extensive Game
How should the game be played?
Solution:
Strategy:
a set of “advisable” strategies, one for each player.
a complete plan of action for every possible decision
node of the game, including nodes that could only
be reached by a mistake at an earlier node.
Player1’s advisable
Player1‘s advisable
strategy in red
Strategy in red
1
1
2
3
Player2’s advisable
strategy in green
2
3
1
2
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22
GAME 1.
Extensive Game
How should the game be played?
If both player’s choose their advisable (prudent) strategies, Player1
will start with 2, Player2 will choose 1, then Player1 will choose 2.
The outcome will be 9 for Player1 (-9 for Player2). If a player makes
a mistake, or deviates, her payoff will be less.
1
1
2
3
2
3
1
2
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22
GAME 1.
Extensive Game
A Clarification:
Rapoport (pp. 49-53) claims Player 1 has 27
strategies.
However, if we consider inconsistent strategies, the
actual number of strategies available to Player 1 is 37 = 2187.
An inconsistent strategy includes actions at decision nodes that
would not be reached by correct implementation at earlier nodes,
i.e., could only be reached by mistake.
Since we can think of a strategy as a set of instructions (or program)
given to an agent or referee (or machine) to implement, a complete
strategy must include instructions for what to do after a mistake is
made. This greatly expands the number of strategies available,
though the essence of Rapoport’s analysis is correct.
Extensive Game
Complete Information:
Players know their own payoffs;
other player(s) payoffs;
history of the game excluding other(s)
current action*
*Actions are simultaneous
1
1
2
3
Information Sets
2
3
1
2
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22
GAME 1.
Matrix Game
T1
Also called
“Normal Form” or
“Strategic Game”
S11
S12
S13
S21
S22
S23
S31
S32
S33
T2
-3 -6
-6
9
-11 -14
-6
9
9 12
-14 17
-11 -14
-14 17
-19 -22
Solution = {S22, T1}
Dominant Strategies
Definition
Dominant Strategy: a strategy that is best no matter what the
opponent(s) choose(s).
T1
T2
T3
T1
T2
T3
S1 -3
0
1
S1 -3
0
-10
S2 -1
5
2
S2 -1
5
2
S3 -2
2
0
S3 -2
-4
0
Dominant Strategies
Definition
Dominant Strategy: a strategy that is best no matter what the
opponent(s) choose(s).
T1
T2
T3
T1
T2
T3
S1 -3
0
1
S1 -3
0
-10
S2 -1
5
2
S2 -1
5
2
S3 -2
2
0
S3 -2
-4
0
Sure Thing Principle: If you have a dominant strategy, use it!
Prudent Strategies
Definitions
T1 T2
T3
S1 -3
1
-20
S2 -1
5
2
S3 -2
-4
15
Prudent Strategy: A prudent strategy for
player i maximizes the
minimum payoff she can
get from playing different
strategies. Such a
strategy is simply
maxsmintP(s,t) for player i.
Player 1’s worst payoffs for
each strategy are in red.
Prudent Strategies
Definitions
T1 T2
T3
S1 -3
1
-20
S2 -1
5
2
S3 -2
-4
15
Prudent Strategy: A prudent strategy for
player i maximizes the
minimum payoff she can
get from playing different
strategies. Such a
strategy is simply
maxsmintP(s,t) for player i.
Player 2’s worst payoffs for
each strategy are in green.
Prudent Strategies
Definitions
T1 T2
T3
S1 -3
1
-20
S2 -1
5
2
S3 -2
-4
15
Prudent Strategy: A prudent strategy for
player i maximizes the
minimum payoff she can
get from playing different
strategies. Such a
strategy is simply
maxsmintP(s,t) for player i.
Saddlepoint:
We call the solution
{S2, T1} a saddlepoint
A set of prudent strategies
(one for each player), s. t.
(s’, t’) is a saddlepoint, iff
maxmin = minmax.
Prudent Strategies
Saddlepoint:
S1 -3
1
-20
S2 -1
5
2
S3 -2
-4
15
A set of prudent strategies
(one for each player), s. t.
(s’, t’) is a saddlepoint, iff
maxmin = minmax.
Mixed Strategies
Player 1
Left
Player 1 hides a button
in his Left or Right hand.
Right
Player 2
L
-2
R
4
L
2
R
-1
GAME 2: Button-Button
Player 2 observes
Player 1’s choice and
then picks either Left or
Right.
Draw the game in matrix
form.
Mixed Strategies
Player 1
Left
Player 1 has 2 strategies;
Player 2 has 4 strategies:
Right
Player 2
L
-2
R
4
L
2
R
-1
GAME 2: Button-Button
LL
RR
LR
RL
L
-2
4
-2
4
R
2
-1
-1
2
Mixed Strategies
Player 1
Left
The game can be solve by
backwards-induction.
Player 2 will …
Right
Player 2
L
-2
R
4
L
2
R
-1
GAME 2: Button-Button
LL
RR
LR
RL
L
-2
4
-2
4
R
2
-1
-1
2
Mixed Strategies
Player 1
Left
The game can be solve by
backwards-induction.
… therefore, Player 1 will:
Right
Player 2
L
-2
R
4
L
2
R
-1
GAME 2: Button-Button
LL
RR
LR
RL
L
-2
4
-2
4
R
2
-1
-1
2
Mixed Strategies
Player 1
Left
Right
L
L
R
-2
4
2
-1
Player 2
L
-2
R
4
L
2
R
-1
GAME 2: Button-Button
R
What would happen if
Player 2 cannot observe
Player 1’s choice?
Solving the Zero-sum Game
L
R
L
-2
4
R
2
-1
Definition
Mixed Strategy: A mixed strategy for
player i is a probability
distribution over all
strategies available to
player i.
Let
(p, 1-p) = prob. Player I chooses L, R.
(q, 1-q) = prob. Player 2 chooses L, R.
GAME 2.
Solving the Zero-sum Game
L
Then Player 1’s expected payoffs are:
R
L
-2
4
(p)
R
2
-1
(1-p)
(q)
(1-q)
EP(L) = -2(p) + 2(1-p) = 2 – 4p
EP(R) = 4(p) – 1(1-p) = 5p – 1
4
EP
EP(R) = 5p – 1
2
0
-1
GAME 2.
p*=1/3
1
p
-2
EP(L) = 2 – 4p
Solving the Zero-sum Game
L
L
-2
Player 2’s expected payoffs are:
R
4
(p)
EP(L) = 2(q) – 4(1-q) = 6q – 4
EP(R) = -2(q) + 1(1-q) = -3q + 1
EP(L) = EP(R)
R
2
-1
(q)
(1-q)
GAME 2.
(1-p)
=>
q* = 5/9
Solving the Zero-sum Game
Player 1
Player 2
EP(L) = -2(p) + 2(1-p) = 2 – 4p
EP(R) = 4(p) – 1(1-p) = 5p – 1
EP(L) = 2(q) – 4(1-q) = 6q – 4
EP(R) = -2(q) + 1(1-q) = -3q + 1
-EP2
EP1
4
-4
2
0
-1
-2
2/3 = EP1* = - EP2* =-2/3
p
p*=1/3
-2
This is the
Value
of the game.
2
q*= 5/9
1 q
2
Solving the Zero-sum Game
L
Then Player 1’s expected payoffs are:
R
L
-2
4
(p)
R
2
-1
(1-p)
(q)
(1-q)
GAME 3.
(Security) Value:
the expected
EP(T1) = -2(p) + 2(1-p)
payoff when both (all)
players play
EP(T ) = 4(p) – 1(1-p)
prudent strategies. 2
EP(T1) = EP(T2)
=> p* = 1/3
Any deviation by an opponent leads
to anPlayer
equal2’s
or greater
payoff.
And
expected
payoffs are:
(V)alue = 2/3
The Minimax Theorem
Von Neumann (1928)
Every zero sum game has a saddlepoint (in pure or
mixed strategies), s.t., there exists a unique value,
i.e., an outcome of the game where
maxmin = minmax.