This document provides an introduction to game theory and how to describe games using matrices and tree diagrams. It defines what constitutes a game, including the key elements of players, their options/moves, possible outcomes, and payoffs. Games can be zero-sum, constant-sum, or variable-sum depending on whether the total payoffs equal zero, remain constant, or vary. Matrix tables are used to describe games like Rock-Paper-Scissors and Matching Pennies by listing the options for each player and their payoffs. Tree diagrams depict games involving sequential moves rather than simultaneous choices. The concept of a dominant strategy is also introduced.
Game Balance 3: Player Equality and FairnessMarc Miquel
In this presentation we introduce the game balance type "player equality and fairness". It is essential so the players do not feel the game is unworthy of playing. All the players must feel they are given the chances to win.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
In this presentation we introduce the game balance type 'sustained uncertainty'. Uncertainty is usually understood as related to randomness and difficulty. It is essential to keep the game interesting to the user.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
Game Balance 3: Player Equality and FairnessMarc Miquel
In this presentation we introduce the game balance type "player equality and fairness". It is essential so the players do not feel the game is unworthy of playing. All the players must feel they are given the chances to win.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
In this presentation we introduce the game balance type 'sustained uncertainty'. Uncertainty is usually understood as related to randomness and difficulty. It is essential to keep the game interesting to the user.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
In this presentation we introduce the game balance "interesting strategies". It is especially important as games with a single dominant strategy are boring. No strategy must be much better than others and without drawbacks.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
Term: 2017-2018 FALL SEMESTER,
Course Name: DECISION THEORY AND ANALYSIS
Department: Industrial Engineering
University: Sakarya University
Lecturer: Halil İbrahim Demir (hidemir.sakarya.edu.tr)
Presenter: Caner Erden (cerden.sakarya.edu.tr)
In this presentation we introduce the concept game balance, its different types, and the most useful methods to study it.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
The importance of symmetrical and asymmetrical balance in the development of games. A video game can be represented in a Excel sheet using numbers only.
This presentation is an attempt to introduce Game Theory in one session. It's suitable for undergraduates. In practice, it's best used as a taster since only a portion of the material can be covered in an hour - topics can be chosen according to the interests of the class.
The main reference source used was 'Games, Theory and Applications' by L.C.Thomas. Further notes available at: http://bit.ly/nW6ULD
Topic includes:
Fairness
Challenge versus success
Meaningful choices
Skill vs chance
Head vs hands
Competition vs cooperation
Short vs long
Rewards
Punishment
Freedom vs controlled experiences
Simple vs complex
Detail vs imagination
Topic includes:
What is game balancing?
Definitions of game balance
Gamification
Games design elements and principles
PvP and PvE
Corporative video games
Practice Exercise
Game balance principles
Importance of game balance
Goals of game balance
In this presentation we introduce the game balance "interesting strategies". It is especially important as games with a single dominant strategy are boring. No strategy must be much better than others and without drawbacks.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
Term: 2017-2018 FALL SEMESTER,
Course Name: DECISION THEORY AND ANALYSIS
Department: Industrial Engineering
University: Sakarya University
Lecturer: Halil İbrahim Demir (hidemir.sakarya.edu.tr)
Presenter: Caner Erden (cerden.sakarya.edu.tr)
In this presentation we introduce the concept game balance, its different types, and the most useful methods to study it.
These slides were prepared by Dr. Marc Miquel. All the materials used in them are referenced to their authors.
The importance of symmetrical and asymmetrical balance in the development of games. A video game can be represented in a Excel sheet using numbers only.
This presentation is an attempt to introduce Game Theory in one session. It's suitable for undergraduates. In practice, it's best used as a taster since only a portion of the material can be covered in an hour - topics can be chosen according to the interests of the class.
The main reference source used was 'Games, Theory and Applications' by L.C.Thomas. Further notes available at: http://bit.ly/nW6ULD
Topic includes:
Fairness
Challenge versus success
Meaningful choices
Skill vs chance
Head vs hands
Competition vs cooperation
Short vs long
Rewards
Punishment
Freedom vs controlled experiences
Simple vs complex
Detail vs imagination
Topic includes:
What is game balancing?
Definitions of game balance
Gamification
Games design elements and principles
PvP and PvE
Corporative video games
Practice Exercise
Game balance principles
Importance of game balance
Goals of game balance
A discussion of basic concepts from game theory, an incredibly useful lemma concerning auctions from mechanism design, and a discussion of TFNP, an interesting complexity class which captures search problems where an answer is guaranteed to exist, such as the problem of finding Nash equilibria in games
2013.05 Games We Play: Payoffs & Chaos MonkeysAllison Miller
Expansion on application of game theory & behavioral analytics to information security and risk management. New concepts include some ideas from coalitional game theory, i.e. not just individual actors but teams.
Students should be able to:
Use simple game theory to illustrate the interdependence that exists in oligopolistic markets
Understanding the prisoners’ dilemma and a simple two firm/two outcome model. Students should analyse the advantages/disadvantages of being a first mover
Students will not be expected to have an understanding of the Nash Equilibrium
2. What are ‘Games?’
• In “game theory,” a ‘GAME’ is an
interaction of decision-makers.
– The Key idea is that players make
decisions that affect one another.
• Ingredients of a game:
1. The players
2. Their options (i.e. possible ‘moves’)
3. Possible outcomes
4. ‘Payoffs’- (i.e. players preferences
among those outcomes)
3. What are ‘Games?’
• A ‘Game’ in this sense includes
games like chess, tic-tac-toe,
football, basketball, etc.
• A ‘game’ as defined also
includes any real-life situation
in which our decisions
influence one another.
– (Remember the definition of
sociology?)
4. Describing Games
• Remember, a game is simply a
situation of interactive
decisions. We can describe
these interactive situations:
1. Verbally
2. Using a matrix (= table)
3. Using a Tree diagram
5. Matrix Descriptions
Rock, Paper, Scissors
STEP 1: Write down the options
for both players in a table.
– Player 1 = row chooser
– Player 2 = column chooser
ROCK PAPER SCISSORS
ROCK
PAPER
SCISSORS
6. Matrix Descriptions
Rock, Paper, Scissors
STEP 2: Write down the ‘payoffs’ (i.e. preferences) for
each possible joint outcome.
– Below I use numbers, +1 to indicate a win, -1, to
indicate a loss, and 0 to indicate a draw.
– Note that there are two different payoffs!
ROCK PAPER SCISSORS
ROCK 0,0 -1, +1 +1, -1
PAPER +1, -1 0, 0 -1, +1
SCISSORS -1, +1 +1, -1 0,0
PLAYER 1
PLAYER 2
7. Matrix Descriptions
Rock, Paper, Scissors
• By convention, the first number is the payoff to
Player 1 (the row chooser). The second number is
the payoff to Player 2 (the column chooser).
– If you only see one number, it is always from the point
of view of Player 1.
ROCK PAPER SCISSORS
ROCK 0,0 -1, +1 +1, -1
PAPER +1, -1 0, 0 -1, +1
SCISSORS -1, +1 +1, -1 0,0
PLAYER 1
PLAYER 2
8. Matrix Descriptions
• Notice that:
1. Players make their moves simultaneously ( they
do not take turns), and also that,
2. R…P…S… is depicted as a ZERO-SUM GAME.
– “Zero-sum” refers to a situation in which the
gains of one player are exactly offset by the
losses of another player. If the total gains of the
participants are added up, and the total losses
are subtracted, they will sum to zero.
• TOTAL GAINS = TOTAL LOSSES
9. Zero-sum
• In a zero-sum game, one person’s
gain comes at the expense of
another person’s loss.
• Example: Imagine a pizza of fixed
quantity. If you eat one more slice
than I do, I necessarily eat one slice
less! More for you = Less for me.
• Example: A thief becomes richer
by stealing from others, but the
total amount of wealth remains the
same.
10. Zero-sum
• Rule: a game is zero-sum if payoffs sum to
ZERO under all circumstances.
– For example, Player 1 chooses Rock and Player 2
chooses Scissors. The aggregate payoff is:
1 – 1 = 0.
ROCK PAPER SCISSORS
ROCK 0,0 -1, +1
+1, -1
PAPER +1, -1 0, 0 -1, +1
SCISSORS -1, +1 +1, -1 0,0
11. Zero-sum
• Example: ‘Matching Pennies’
– Rules: In this two-person game, each player takes a penny
and places it either heads-up or tails-up and covers it so
the other player cannot see it. Both players’ pennies are
then uncovered simultaneously. Player 1 is called
Matchmaker and gets both pennies if they show the same
face (heads or tails). Player 2 is called Variety-seeker and
gets both pennies if they show opposite faces (one heads,
the other tails).
HEADS TAILS
HEADS +1, -1 -1, +1
TAILS -1, +1 +1, -1
Matchmaker
Variety-Seeker
12. Constant-sum and Variable-sum
• Not all games are zero-sum games!
1. A situation in which the total payoffs are fixed
and never change, but do not necessarily equal
zero, is called a constant-sum game.
– Note: Zero-sum games are a kind of constant-sum
game in which the constant-sum is zero.
2. Variable-sum games are those in which the sum
of all payoffs changes depending on the choices
of the players! The game prisoner’s dilemma is
a classic example of this! (You will have to show
this yourself)
13. Tree Diagrams
• Tree diagrams (aka ‘decision-trees’) are useful
depictions of situations involving sequential
turn-taking rather than simultaneous moves.
• Asking Boss for a Raise?
Employee
0,0
Boss
2, -2
-1, 0
15. Dominant Strategy
• In Game Theory, a player’s dominant strategy
is the choice that always leads to a higher
payoff, regardless of what the other player(s)
choose.
– Not all games have a dominant strategy, and
games may exist in which one player has a
dominant strategy but not the other.
– In the game prisoner’s dilemma, both players
have a dominant strategy. Can you determine
which choice dominates the others?