Zero-sum Games
• 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 Game1. 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.
4. Information: What players know (believe) when choosing actions.
ZERO-SUM
The Essentials of a Game
4. Information: What players know (believe) when choosing actions.
Perfect Information: Players know
• their own payoffs • other player(s) payoffs • the history of the game, including other(s) current action*
*Actions are sequential (e.g., chess, tic-tac-toe).
Common Knowledge
Extensive GamePlayer 1 chooses a = {1, 2 or 3} Player 2 b = {1 or 2} Player 1 c = {1, 2 or 3}
Payoffs = a2 + b2 + c2 if /4 leaves remainder of 0 or 1. -(a2 + b2 + c2) if /4 leaves remainder of 2 or 3. Player1’sdecision nodes
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.
“Square the Diagonal”(Rapoport: 48-9)
Player 2’sdecision nodes
1 32
1 21
23
Extensive GameHow should the game be played?Solution: a set of “advisable” strategies, one for each player.Strategy: 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 Strategy in red
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.
1 32
1 21
23
Start at the final decision nodes (in red) Backwards-induction
Extensive GameHow should the game be played?Solution: a set of “advisable” strategies, one for each player.Strategy: 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 Strategy in red
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.
Player2’s advisable strategy in green
1 32
1 21
23
Player1’s advisable strategy in red
Extensive GameHow 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.
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.
1 32
1 21
23
Extensive GameA 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 GameComplete Information: Players know their own payoffs;
other player(s) payoffs; history of the game excluding other(s) current action*
*Actions are simultaneous
-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.
1 32
1 21
23
Information Sets
Matrix Game
-3 -6-6 9
-11 -14-6 99 12
-14 17-11 -14-14 17-19 -22
T1 T2
Also called “Normal Form” or “Strategic Game”
Solution = {S22, T1}
S11
S12
S13
S21
S22
S23
S31
S32
S33
Dominant StrategiesDefinition
Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).
T1 T2 T3 T1 T2 T3
-3 0 -10
-1 5 2
-2 -4 0
-3 0 1
-1 5 2
-2 2 0
S1
S2
S3
S1
S2
S3
Dominant StrategiesDefinition
Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).
T1 T2 T3 T1 T2 T3
Sure Thing Principle: If you have a dominant strategy, use it!
-3 0 -10
-1 5 2
-2 -4 0
-3 0 1
-1 5 2
-2 2 0
S1
S2
S3
S1
S2
S3
Prudent Strategies
T1 T2 T3
Player 1’s worst payoffs for each strategy are in red.
-3 1 -20
-1 5 2
-2 -4 15
S1
S2
S3
Definitions
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.
Prudent Strategies
T1 T2 T3
Player 2’s worst payoffs for each strategy are in green.
-3 1 -20
-1 5 2
-2 -4 15
S1
S2
S3
Definitions
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.
Prudent Strategies
T1 T2 T3
-3 1 -20
-1 5 2
-2 -4 15
S1
S2
S3
Definitions
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: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff maxmin = minmax.
We call the solution {S2, T1} a saddlepoint
Prudent Strategies
-3 1 -20
-1 5 2
-2 -4 15
S1
S2
S3
Saddlepoint: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff
maxmin = minmax.
Mixed Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
Player 1 hides a button in his Left or Right hand.
Player 2 observes Player 1’s choice and then picks either Left or Right.
Draw the game in matrix form.
Mixed Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
Player 1 has 2 strategies;Player 2 has 4 strategies:
-2 4 -2 4
2 -1 -1 2
L
R
LL RR LR RL
Mixed Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
The game can be solve by backwards-induction. Player 2 will …
-2 4 -2 4
2 -1 -1 2
L
R
LL RR LR RL
Mixed Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
The game can be solve by backwards-induction. … therefore, Player 1 will:
-2 4 -2 4
2 -1 -1 2
L
R
LL RR LR RL
Mixed Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
-2 4
2 -1
L R
L
R
GAME 2: Button-Button
What would happen if Player 2 cannot observe Player 1’s choice?
Solving the Zero-sum Game
GAME 2.
-2 4
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.
L R
L
R
Solving the Zero-sum Game
GAME 2.
-2 4
2 -1
Then Player 1’s expected payoffs are: EP(L) = -2(p) + 2(1-p) = 2 – 4p EP(R) = 4(p) – 1(1-p) = 5p – 1
L R
L
R
(p)
(1-p)
(q) (1-q)0 1 p
EP(L) = 2 – 4p
EP(R) = 5p – 1
EP
p*=1/3
2
-1
4
-2
Solving the Zero-sum Game
GAME 2.
-2 4
2 -1
Player 2’s expected payoffs are:
EP(L) = 2(q) – 4(1-q) = 6q – 4 EP(R) = -2(q) + 1(1-q) = -3q + 1
EP(L) = EP(R) => q* = 5/9
L R
L
R
(p)
(1-p)
(q) (1-q)
Solving the Zero-sum GamePlayer 1
EP(L) = -2(p) + 2(1-p) = 2 – 4p EP(R) = 4(p) – 1(1-p) = 5p – 1
0 p 1 q
-EP2
p*=1/3
2
-1
4
-2
2/3 = EP1* = - EP2* =-2/3
This is the
Value
of the game.
EP1
-4
2
-2
2
q*= 5/9
Player 2
EP(L) = 2(q) – 4(1-q) = 6q – 4 EP(R) = -2(q) + 1(1-q) = -3q + 1
Solving the Zero-sum Game
GAME 3.
-2 4
2 -1
Then Player 1’s expected payoffs are:
EP(T1) = -2(p) + 2(1-p) EP(T2) = 4(p) – 1(1-p)
EP(T1) = EP(T2) => p* = 1/3
And Player 2’s expected payoffs are:
(V)alue = 2/3
L R
L
R
(p)
(1-p)
(q) (1-q)
(Security) Value: the expected payoff when both (all) players play prudent strategies.
Any deviation by an opponent leads to an equal or greater payoff.
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.