What are ‘Games?’• In “game theory,” a ‘GAME’ is aninteraction of decision-makers.– The Key idea is that players makedecisions that affect one another.• Ingredients of a game:1. The players2. Their options (i.e. possible ‘moves’)3. Possible outcomes4. ‘Payoffs’- (i.e. players preferencesamong those outcomes)
What are ‘Games?’• A ‘Game’ in this sense includesgames like chess, tic-tac-toe,football, basketball, etc.• A ‘game’ as defined alsoincludes any real-life situationin which our decisionsinfluence one another.– (Remember the definition ofsociology?)
Describing Games• Remember, a game is simply asituation of interactivedecisions. We can describethese interactive situations:1. Verbally2. Using a matrix (= table)3. Using a Tree diagram
Matrix DescriptionsRock, Paper, ScissorsSTEP 1: Write down the optionsfor both players in a table.– Player 1 = row chooser– Player 2 = column chooserROCK PAPER SCISSORSROCKPAPERSCISSORS
Matrix DescriptionsRock, Paper, ScissorsSTEP 2: Write down the ‘payoffs’ (i.e. preferences) foreach possible joint outcome.– Below I use numbers, +1 to indicate a win, -1, toindicate a loss, and 0 to indicate a draw.– Note that there are two different payoffs!ROCK PAPER SCISSORSROCK 0,0 -1, +1 +1, -1PAPER +1, -1 0, 0 -1, +1SCISSORS -1, +1 +1, -1 0,0PLAYER 1PLAYER 2
Matrix DescriptionsRock, Paper, Scissors• By convention, the first number is the payoff toPlayer 1 (the row chooser). The second number isthe payoff to Player 2 (the column chooser).– If you only see one number, it is always from the pointof view of Player 1.ROCK PAPER SCISSORSROCK 0,0 -1, +1 +1, -1PAPER +1, -1 0, 0 -1, +1SCISSORS -1, +1 +1, -1 0,0PLAYER 1PLAYER 2
Matrix Descriptions• Notice that:1. Players make their moves simultaneously ( theydo 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 thegains of one player are exactly offset by thelosses of another player. If the total gains of theparticipants are added up, and the total lossesare subtracted, they will sum to zero.• TOTAL GAINS = TOTAL LOSSES
Zero-sum• In a zero-sum game, one person’sgain comes at the expense ofanother person’s loss.• Example: Imagine a pizza of fixedquantity. If you eat one more slicethan I do, I necessarily eat one sliceless! More for you = Less for me.• Example: A thief becomes richerby stealing from others, but thetotal amount of wealth remains thesame.
Zero-sum• Rule: a game is zero-sum if payoffs sum toZERO under all circumstances.– For example, Player 1 chooses Rock and Player 2chooses Scissors. The aggregate payoff is:1 – 1 = 0.ROCK PAPER SCISSORSROCK 0,0 -1, +1+1, -1PAPER +1, -1 0, 0 -1, +1SCISSORS -1, +1 +1, -1 0,0
Zero-sum• Example: ‘Matching Pennies’– Rules: In this two-person game, each player takes a pennyand places it either heads-up or tails-up and covers it sothe other player cannot see it. Both players’ pennies arethen uncovered simultaneously. Player 1 is calledMatchmaker and gets both pennies if they show the sameface (heads or tails). Player 2 is called Variety-seeker andgets both pennies if they show opposite faces (one heads,the other tails).HEADS TAILSHEADS +1, -1 -1, +1TAILS -1, +1 +1, -1MatchmakerVariety-Seeker
Constant-sum and Variable-sum• Not all games are zero-sum games!1. A situation in which the total payoffs are fixedand never change, but do not necessarily equalzero, is called a constant-sum game.– Note: Zero-sum games are a kind of constant-sumgame in which the constant-sum is zero.2. Variable-sum games are those in which the sumof all payoffs changes depending on the choicesof the players! The game prisoner’s dilemma isa classic example of this! (You will have to showthis yourself)
Tree Diagrams• Tree diagrams (aka ‘decision-trees’) are usefuldepictions of situations involving sequentialturn-taking rather than simultaneous moves.• Asking Boss for a Raise?Employee0,0Boss2, -2-1, 0
Dominant Strategy• In Game Theory, a player’s dominant strategyis the choice that always leads to a higherpayoff, regardless of what the other player(s)choose.– Not all games have a dominant strategy, andgames may exist in which one player has adominant strategy but not the other.– In the game prisoner’s dilemma, both playershave a dominant strategy. Can you determinewhich choice dominates the others?