Guillermo Sabbioni
Game Theory (ECO4400)
Book: Games of Strategy, by Dixit & Skeath
Chapter 4: Simultaneous-move games with pure strategies: discrete strategies.
If moving at the same time: you don’t know what the other is choosing (simultaneous
move games). Other name: imperfect information games. Examples: voting, soccer
kicker and goalie (can’t afford to wait). Each player must figure out what others are doing
when the others are trying to figure out what this player is doing (circularity).
1. Depicting simultaneous-move games with discrete strategies
Strategy=action in simultaneous games (players choose only once). True either for
discrete or continuous choices. Game: described by a matrix or payoff table (see Fig 4.1).
If more than 2 players: other dimension besides row and column (page, for ex). By
convention: the first payoff corresponds to the row player. If the payoff for one player is
the negative of the payoff for the other player: zero-sum games (or constant-sum
games). In that case: only list one player’s payoffs (usually Row). See fig 4.2.
2. Nash equilibrium
Best Response: what is the best you can do given what your rival does (mental exercise).
See fig 4.1 and focus on Low, Middle. Nobody gains by deviating. Both strategies are
best responses. That is a Nash Equilibrium. Recall: players make their choices
independently (non cooperative game).
Definition: a NE is a list of strategies (1 for each player) such that no player can get a
better payoff by switching to other strategy while all the other players adhere to the
strategies specified for them in that list.
a. Some further explanation
Why not High, Left? Row wants to deviate to Bottom. Also, no need for NE payoffs to be
strictly better than the other available choices. See fig 4.3. Also, no need for a NE to be
jointly best for both players. See also fig 4.2: Short-Pass by Offense and Pass by Defense
is a NE. How to find a NE? Cell by cell inspection is possible (tedious?).
b. NE as a system of beliefs and choices
Paradox: How do you respond if you don’t see the other’s choice? You put yourself in
the other’s shoes: think about what he is thinking… circle. NE is the culmination of this
process of thinking about thinking, where each player correctly figures out the others’
choice. Best word found so far: belief. Then, redefinition: a NE is a list of strategies such
that (1) each player has correct beliefs about the strategies of the others and (2) the
strategy of each player is the best, given her beliefs about the strategies of the others.
Then: concept of best response is not logically flawed.
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Guillermo Sabbioni
3. Dominance
Search for a NE is simplified if the player/s have dominant strategies. Dominant
strategy: one that is uniformly better than all the others. Example: Prisoner’s dilemma. If
neither confesses: 3 years each for a minor charge. If only one confesses: only 1 year (for
cooperating with the police) while the other gets 25 years (only one guilty). If both
confess: both 10 years. Confess is a dominant strategy: better payoff regardless of the
rival’s choice. Equivalently: Deny is a dominated strategy. See fig 4.4. If you have a
dominant strategy: choose it.
a. Both players have dominant strategies
Easy: both players choose their dominant strategy. The (unique in this case) NE arises
automatically. (Confess, Confess) is the NE.
Three characteristics of a Prisoners’ dilemma game: Each player has 2 strategies
(cooperate or not), each player has a dominant strategy and the equilibrium from the
dominant strategies is worse for both than the non equilibrium where each player
cooperates (plays the dominated strategy).
The key: cooperation could be achieved (better payoffs for both players) if there was a
way to guarantee that nobody will cheat (very difficult).
b. One player has a dominant strategy
The player with a dominant strategy will use it, and the rival can safely choose his best
response to that. See fig 4.5. Payoffs depend on the underlying macroeconomic situation.
FED does not have a dominant strategy. But Congress does: Deficit. FED then chooses
the BR to FED dominant strategy. NE: (Deficit, High rates). Notice that (Balance, Low)
is jointly better. But not a NE: Congress would deviate and play Deficit.
c. Successive elimination of dominated strategies
If more than 2 strategies available, a strategy can be eliminated after the elimination of a
dominated strategy for another player. The game gets smaller. If the process ends in a
unique outcome, the game is dominance solvable (by the iterated elimination of
dominated strategies). See fig 4.1 again. We get to (Low, Middle) by that process. It
does not always lead to a unique cell. But it usually helps reducing the size of the game.
Also: easy if there are no ties. See fig 4.3. Can’t get to the unique NE (game is not
dominance solvable). Careful: elimination of weakly dominated strategies can get rid of
NE. See fig 4.6. Only (Down, Right) remains (Two NE are ruled out).
4. Best response analysis
Another way to look for NE: for each possible choice of the other players, what is the
best choice for this player? Very useful if no dominant strategies exist. See fig 4.7. If
there is a dominant strategy: that one is the BR for all possible choices of the other
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players. Both payoffs circled? NE is there. If the BR analysis does not find a NE, the
game does not have a NE (in pure strategies!!!!!).
5. The minimax method for zero-sum games
Minimax method: It works only for zero-sum games. It uses the innate conflict of those
games: one player wins only when the other loses (not necessarily true in non zero-sum
games). You should choose the best of your worst payoffs. It looks pessimistic, but it is
actually fine. Since Column wants to minimize Row’s payoffs (from the inherent conflict
between both), Column chooses the lowest payoff (picking a column) for each row that
Row may choose. Then Row should pick the row that yields the highest payoff given
what Column would choose (to minimize Row’s payoff). Row chooses to maximin.
Similarly, Column chooses to minimax. See fig 4.8. If the method fails: no NE in pure
strategies (again).
6. Three players
Analyze the garden game if played simultaneously. If there are three players, add a
“page” player (Talia in this case). See fig 4.9. Everything else is the same. Very careful
when looking for dominant strategies! (it is easy to screw up). Emily: dominant strategy
Don’t (for both of Talia’s choices!). Nina: dominant strategy Don’t (for both of Talia’s
choices again! Don’t forget!). Checking for Talia: keep Nina and Emily fixed and
evaluate across pages (Up,Left on the right table vs Up,Left on the left table, for
example). Talia also has a dominant strategy: Don’t. Hence: NE is don’t, don’t, don’t by
all players. Using cell by cell inspection or BR analysis yields the same result. See fig
4.10.
7. Multiple equilibria in pure strategies
Some games: multiple NE. A special group of these games with multiple NE:
coordination games. Feature: some common interests. Coordination would be great, but
it may fail due to the non cooperative characteristic (you choose individually). See fig
4.11. This is pure coordination: which NE is achieved does not matter, as long as they
get to one NE. If one of the cafes seems like a natural choice: focal point. Usually: not an
obvious answer. What if one NE is preferred by both? See fig 4.12. However, that does
not guarantee that the best NE will arise. Only if each has assurance that the other is
choosing the appropriate action: assurance games. In real life, it should be easy to
achieve. What if both players want to meet, but prefer different cafes? See fig 4.13.
Structure of a battle of the sexes game (originally: a boxing match or ballet). Greater risk
of coordination here (vs pure coordination and assurance games). Preferences over both
NE are now in conflict (not in pure coordination or assurance). Being tough or nice?
Final example: chicken game. See fig 4.14. Each player: Tough and Weak strategy.
There are 2 NE: both entail one player being tough and the other weak. Each player
prefers a different NE. If both are tough: pretty bad outcome (“huge crash”). Solution: try
to intimidate, have a reputation, play it repeatedly, etc (we will see that later). It is a kind
of battle of the sexes (conflict about which NE is desired).
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Guillermo Sabbioni
8. No equilibrium in pure strategies
Not all games have a NE in pure strategies. Example: single point in a tennis match. See
fig 4.15. No dominant strategy, cell by cell inspection fails as well as BR analysis.
Lesson: don’t choose always the same strategy. Randomize, but in an unpredictable
fashion. Otherwise: the rival will exploit this pattern. That is called a mixed strategy.
SUGGESTED EXERCISES: 2, 3, 4, 5 and 7.
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