Download Game Theory: Matrix Games and Nash Equilibria and more Lecture notes Game Theory in PDF only on Docsity!
Game Theory
Game Theory (GT) is the study of strategic interdependence. The typical “game” consists of players, actions, strategies and payoffs. The standard modes of analysis are once-played games in either
(i) matrix or strategic form where players’ choice of actions are made simultaneously,
or
(ii) extensive form where choice of actions are made sequentially.
1 Analysis of (finite) Matrix Games
One of the best known games is
Prisoner’s Dilemma (PD)
Player 2 Keep Quiet Confess Player 1 Keep Quiet (-1,-1) (-12,0) Confess (0,-12) (-8,-8)
The solution is 〈 confess, confess 〉. (See IESDS). Why don’t the players coordinate to get 〈 Keep Quiet, Keep Quiet 〉?
The exact payoffs are irrelevant, the game can also be represented by the order of players’ preferences - most preferred (p1) to least preferred (p4) :
Player 2 Keep Quiet Confess Player 1 Keep Quiet (p2,p2) (p4,p1) Confess (p1,p4) (p3,p3)
Other examples of PD-like games are
- War with strategies Defend, Attack respectively.
- Arms Race with strategies Pass, Build respectively.
- Free Trade/ Protection with strategies No Tax, Tax respectively.
- Advertising with strategies No Ads, Ads respectively.
Deadlock is another game ( success is to fail!):
Player 2 Try Fail
Player 1 Try (0,0) (-1,1) Fail (1,-1) (0,0)
Using preferences, we can consider the more general version of deadlock
Player 2 Left Right Player 1 Up (p2,p2) (p1,p4) Down (p4,p1) (p3,p3)
The solution is 〈 Up, Left 〉. (See IESDS). Neither player gets hir first choice unless the other makes a mistake.
1.1 Strict Dominance
Strategy X strictly dominates strategy Y for a player if X gives a bigger (more preferred) payoff than Y no matter what the other players do. Players never rationally choose strictly dominated strategies.
Reduce the matrix by Iterated Elimination of Strictly Dominated Strategies (IESDS) - see PD and Deadlock above. The order of elimination is irrelevant. Another example is the Dance Club game:
Boon docks Salsa Hip Hop Downtown Salsa (80,0) (60,40) Hip Hop (40,60) (40,0)
The solution is 〈 Salsa, Hip Hop 〉. Salsa dominates Hip Hop for Club Downtown, then Boonies choses Hip Hop.
Some more examples
Player 2 Left Centre Right
Player 1
Up (13,3) (1,4) (7, 3) Middle (4,1) (3,3) (6,2) Down (-1,9) (2,8) (8,-1)
Denoting strict dominance by >, then in the order given C > R, M > D, C > L, M > U. Thus the solution is 〈 Middle, Centre 〉.
Cournot Duopoly game. Firm 1 can produce i units at a cost of e1 each. Similarly firm 2 can produce j units at a cost of e1 each. The units sell on the market at a price of e[8 − 2(i + j)]+^ each, where [.]+^ is the positive part of [.] The payoff to firm 1 is the profit gained which is e[8 − 2(i + j)]+i − i. Similarly the payoff to firm 2 is e[8 − 2(i + j)]+j − j. The game matrix is
For this game there is no SDS nor WDS.
A Nash equilibrium (NE) is a set of strategies, one for each player, from which there is no incentive for any one player to deviate if all the other players play these strategies, i.e. no player can gain by changing, also called a “No regrets” choice. The Best Response of a player to another player’s choice of strategy is the strategy that gives the largest or best payoff. We’ll denote this by placing an ∗^ beside the payoff, e.g. in the stag game above , “stag” with associated payoff 3 is the best response of player 1 to player 2 playing “stag”. Hence if both parts of a payoff pair have asterisks beside them, this must be a pure strategy Nash equilibrium (PSNE) - a pair of strategies where both players are playing deterministic strategies as opposed to a mixed strategy Nash equilibrium (MSNE), where players are randomly mixing between the strategies available to them.
Thus in the stag game, the PSNE solutions are 〈 stag, stag 〉 and 〈 hare, hare 〉. Notice that without efficient coordination, either solution is possible.
Consider the “Good buddies prisoner’s dilemma” game:
Player 2 Keep Quiet Confess
Player 1 Keep Quiet (p1,p1) (p4,p2) Confess (p2,p4) (p3,p3)
This is identical to the stag hunt game.
An alternative definition of a NE is “mutual best response”. Some further examples: Traffic Lights(TL)
Car 2 Go Stop Car 1 Go (-5,-5) (1,0) Stop (0,1) (-1,-1)
The PSNE solutions are 〈 go, stop 〉 and 〈 stop, go 〉.
Generals, Armies & Battles. Each general commands 3 armies. No battle occurs if either general puts 0 armies in the field. Otherwise the general with more armies wins the day. With i andj standing for the number of armies of General 1 and General 2 respectively in the battlefield, one possible game matrix is
General 2 j = 0 j = 1 j = 2 j = 3
General 1
i = 0 (0,0) (0,0) (0, 0) (0,0) i = 1 (0,0) (0,0) (-1,1) (-1,1) i = 2 (0,0) (1,-1) (0,0) (-1,1) i = 3 (0,0) (1,-1) (1,-1) (0,0)
What are the PSNE(s)?
1.4 Dominance & NE
If IESDS results in a unique solution then it is a (unique) NE. [proof by appeal to “no regrets”]. IESDS does not remove any NE.
IEWDS may lose NE. It is necessary to check using e.g. Best Responses. If you have a choice eliminate SDS before WDS. Examples:
- The example of Section 1.2 has two NE, each obtained by a different sequence of IEWDS.
- Consider the game
Player 2 Left Right Player 1 Up (2,3) (4,3) Down (3,3) (1,1)
Using IEWDS, L ≥ R, then D > U. Hence the solution is 〈 Down, Left 〉. But using Best Responses, another NE is 〈 Up, Right 〉.
Player 2 Left Centre Right
Player 1
Up (2,2) (4,2) (4, 3) Middle (2,4) (5,5) (7,3) Down (3,4) (3,7) (6,6)
Using IEWDS C ≥ L, then M > U, M > D and C> R. Hence the solution is 〈 Middle, Centre 〉. This is the only NE.
1.5 MSNE
Matching Pennies(MP) is an example of a game with no PSNE.
Player 2 Heads Tails
Player 1 Heads (1,-1) (-1,1) Tails (-1,1) (1,-1)
Other names for this game are
- Goalkeeeper v. Penalty Taker
- Offense v. Defense (American Football)
- Fastball v. Curveball (Baseball)
respectively. To be indifferent between the two requires
3 q + (−2)(1 − q) = (−1)q + 0(1 − q) ⇒ q = 1 / 3
and the value of the payoff is
v 1 = R 1
Up or Down,
Left +
Right
Exercise: Show that the stag hunt game has a MSNE at 〈 1/2 stag + 1/2 hare, 1/2 stag
We can have partial MSNE where (at least) one player has a pure strategy and (at least) one player has a mixed strategy.
Examples of games with MSNE:
- Chicken (aka Snowdrift) :
Player 2 Continue / Stay in car Swerve / Shovel Player 1 Continue / Stay in car (-10,-10) (2,-2) Swerve / Shovel (-2,2) (0,0)
PSNEs occur at 〈 Continue, Swerve 〉 and at 〈 Swerve, Continue 〉. There is also a MSNE at 〈 1/5 continue + 4/5 swerve, 1/5 continue + 4/5 swerve 〉. Show that v 1 = v 2 = − 2 /5.
Her Ballet Fight Him Ballet (1,2) (-1,1) Fight (-1,1) (1,-1)
PSNEs occur at 〈 Ballet, Ballet 〉 and at 〈 Fight, Fight 〉. Show that there is a MSNE at 〈 1/3 ballet + 2/3 fight, 2/3 ballet + 1/3 fight 〉 with v 1 = 2/3 = v 2. Compare and contast this with the payoffs of the PSNEs.
1.6 MSNE and Dominance
A SDS cannot be played with positive probability in a MSNE (otherwise a higher payoff can be obtained by not playing the SDS when the strategy say to play it).
1.7 Strict Dominance in Mixed Strategies
Consider the game:
Player 2 Left Right
Player 1
Up (3,-1) (-1,1) Middle (0,0) (0,0) Down (-1,2) (2,-1)
This game has no SDS dominated by a pure strategy nor any PSNE. If a mixture of two pure strategies dominates another, then that strategy is a SDS. In the above game 1/2 Up + 1/2 Down > Middle. Remove Middle to get
Player 2 Left Right
Player 1 Up (3,-1) (-1,1) Down (-1,2) (2,-1)
Show that a MSNE exists at 〈 (3/5)Up + (2/5)Down, (3/7)Left + (4/7)Right 〉.
Another example:
Player 2 Left Centre Right
Player 1
Up (-3,6) (9,1) (0, 2) Middle (3,-4) (2,4) (4,1) Down (4,7) (3,2) (-3,2)
Using IESDS, we get (1/4)L + (3/4)C > R, D > M, L > C, D > U. Thus the solution is 〈 Down, Left 〉.
1.8 Atypical Matrix Games
Almost all matrix games have an odd number of solutions (Wilson 1971). Examples of non generic games follow. Weak dominance is usually the culprit.
Player 2 Left Right
Player 1 Up (1,1) (0,0) Down (0,0) (0,0)
There are two PSNEs.
