Dominant and Dominated Strategies - Illinoishrtdmrt2/Teaching/GT_2015_19/L3.pdf · Dominant and...

Dominant and Dominated Strategies

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

May 29th, 2015

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Formalizing the Game

2 Dominant and Dominated Strategies

3 Iterated Delation of Strictly Dominated Strategies

4 Iterated Delation of Dominated Strategies

5 Exercises

C. Hurtado (UIUC - Economics) Game Theory

Formalizing the Game

On the Agenda





5 Exercises

C. Hurtado (UIUC - Economics) Game Theory 1 / 22



Up to this point we defined game without been formal. Let me introduce someNotation:

- set of players: I = {1, 2, · · · ,N}- set of actions: ∀i ∈ I, ai ∈ Ai , where each player i has a set of actions Ai .- strategies for each player: ∀i ∈ I, si ∈ Si , where each player i has a set of

pure strategies Si available to him. A strategy is a complete contingent planfor playing the game, which specifies a feasible action of a player’sinformation sets in the game.

- profile of pure strategies: s = (s1, s2, · · · , sN) ∈∏N

i=1 Si .Note: let s−i = (s1, s2, · · · , si−1, si+1, · · · , sN) ∈ S−i , we will denotes = (si , s−i) ∈ (Si , S−i).

- Payoff function: ui :∏N

i=1 Si → R, denoted by ui(si , s−i)




Now we can denote game with pure strategies and complete information in normalform by: ΓN = {I, {Si}i , {ui}i}.What about the games with mix strategies?We have taken it that when a player acts at any information set, hedeterministically picks an action from the set of available actions. But there is nofundamental reason why this has to be case.

DefinitionA mixed strategy for player i is a function σi : Si → [0, 1], which assigns a probabilityσi(si) ≥ 0 to each pure strategy si ∈ Si , satisfying

∑si∈Si

σi(si) = 1.

We denote the set of mixed strategies by ∆(Si).Note that a pure strategy can be viewed as a special case of a mixed strategy inwhich the probability distribution over the elements of Si is degenerate.



Example

Meeting in New York:- Players: Two players, 1 and 2- Rules: The two players can not communicate. They are suppose to meet in NYC

at noon to have lunch but they have not specify where. Each must decide whereto go (only one choice).

- Outcomes: If they meet each other, they enjoy other’s company. Otherwise, theyeat alone.

- Payoffs: They attach a monetary value of 100 USD to other’s company and 0USD to eat alone.

player 2A B C

player 1A 100,100 0,0 0,0B 0,0 100,100 0,0



Example

Meeting in New York:- set of players: I = {1, 2}- set of actions: A1 = {A,B}, and A2 = {A,B,C}- strategies for each player: S1 = A1, and S2 = A2 (Why?)- Payoff function: ui :

∏2i=1 Si → R, denoted by ui(si , s−i)

u(si , s−i) =

{100

0if si = s−i

if si 6= s−i

Player 2- pure strategies: S2 = {A,B,C}. Player 2 has 3 pure strategies.- mixed strategies:∆(S2) = {(σ2

1 , σ22 , σ

23) ∈ R3|σ2

m ≥ 0∀m = 1, 2, 3 and∑3

m=1 σ2m = 1}



On the Agenda





5 Exercises




Now we turn to the central question of game theory: What should be expected toobserve in a game played by rational agents who are fully knowledgeable about thestructure of the game and each others’ rationality?To keep matters simple we initially ignore the possibility that players mightrandomize in their strategy choices.The prisoner’s dilemma:

* Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitaryconfinement with no means of speaking to or exchanging messages with the other.

* The prosecutors do not have enough evidence to convict the pair on the principal charge.They hope to get both sentenced to a year in prison on a lesser charge.

* Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given theopportunity either to: betray the other by testifying that the other committed the crime,or to cooperate with the other by remaining silent.

* Here is the offer:- If A and B each betray the other, each of them serves 2 years in prison- If A betrays B but B remains silent, A will be set free and B will serve 3 years in

prison (and vice versa)- If A and B both remain silent, both of them will only serve 1 year in prison (on the

lesser charge)




Let me put prisoner’s dilemma as a game of trust:

player 2trust cheat

player 1trust 5,5 1,10cheat 10,1 2,2

Observe that regardless of what her opponent does, player i is strictly better offplaying Cheat rather than Trust. This is precisely what is meant by a strictlydominant strategy.Player 2 plays Trust. Player 1 knows that 10 > 5, better to Cheat.Player 2 plays Cheat. Player 1 knows that 2 > 1, better to Cheat.Regardless of the other’s strategies, it is always better to Cheat.Note that both would be better off if they both play trust.Lesson: self-interested behavior in games may not lead to socially optimaloutcomes.




DefinitionA strategy si ∈ Si is a strictly dominant strategy for player i if for alls̃i 6= si and all s−i ∈ S−i , ui(si , s−i) > ui(s̃i , s−i).

A strictly dominant strategy for i uniquely maximizes her payoff for any strategyprofile of all other players.If such a strategy exists, it is highly reasonable to expect a player to play it. In asense, this is a consequence of a player’s ”rationality”.




What about if a strictly dominant strategy doesn’t exist?

player 2a b c

player 1A 5,5 0,10 3,4B 3,0 2,2 4,5

You can easily convince yourself that there are no strictly dominant strategies herefor either player.Notice that regardless of whether Player 1 plays A or B, Player 2 does strictlybetter by playing b rather than a.That is, a is ”strictly dominated” by b.




DefinitionA strategy si ∈ Si is strictly dominated for player i if there exists astrategy s̃i ∈ Si such that for all s−i ∈ S−i , ui(s̃i , s−i) > ui(si , s−i). In thiscase, we say that s̃i strictly dominates si .

In words, s̃i strictly dominates si if it yields a strictly higher payoff regardless ofwhat (pure) strategy rivals use.Note that the definition would also permits us to use mixed strategiesUsing this terminology, we can restate the definition of strictly dominant: Astrategy si is strictly dominant if it strictly dominates all other strategies.It is reasonable that a player will not play a strictly dominated strategy, aconsequence of rationality, again.


Iterated Delation of Strictly Dominated Strategies

On the Agenda





5 Exercises




player 2a b c

player 1A 5,5 0,10 3,4B 3,0 2,2 4,5

We argued that a is strictly dominated (by b) for Player 2; hence rationality ofPlayer 2 dictates she won’t play it.We can push the logic further: if Player 1 knows that Player 2 is rational, heshould realize that Player 2 will not play strategy a.Notice that we are now moving from the rationality of each player to the mutualknowledge of each player’s rationality.Once Player 1 realizes that 2 will not play a and ”deletes” this strategy from thestrategy space, then strategy A becomes strictly dominated by strategy B forPlayer 2.If we iterate the knowledge of rationality once again, then Player 2 realizes that 1will not play A, and hence ”deletes” A.Player 2 should play c. We have arrived at a ”solution”.




DefinitionA game is strict-dominance solvable if iterated deletion of strictlydominated strategies results in a unique strategy profile.

Since in principle we might have to iterate numerous times in order to solve astrict-dominance solvable game, the process can effectively can only be justified bycommon knowledge of rationality.As with strictly dominant strategies, it is also true that most games are notstrict-dominance solvable.You might worry whether the order in which we delete strategies iterativelymatters. Insofar as we are working with strictly dominated strategies so far, it doesnot.


Iterated Delation of Dominated Strategies

On the Agenda





5 Exercises




DefinitionA strategy si ∈ Si is a weakly dominant strategy for player i if for alls̃i 6= si and all s−i ∈ S−i , ui(si , s−i) ≥ ui(s̃i , s−i), and for at least onechoice of s−i the inequality is strict.

DefinitionA strategy si ∈ Si is weakly dominated for player i if there exists a strategys̃i ∈ Si such that for all s−i ∈ S−i , ui(s̃i , s−i) ≥ ui(si , s−i), and for at leastone choice of s−i the inequality is strict. In this case, we say that s̃i weaklydominates si .

DefinitionA game is weakly-dominance solvable if iterated deletion of weaklydominated strategies results in a unique strategy profile.




Using this terminology, we can restate the definition of weakly dominant: Astrategy si is weakly dominant if it weakly dominates all other strategies.You might worry whether the order in which we delete strategies iterativelymatters. Delation of dominated strategies could leave to different outcomes.

P2L R

U 5,1 4,0P1 M 6,0 3,1

D 6,4 4,4

P2L R

P1U 5,1 4,0D 6,4 4,4

P2L R

P1M 6,0 3,1D 6,4 4,4


Exercises

On the Agenda





5 Exercises


Exercises

ExercisesExercise 1. Prove that a player can have at most one strictly dominant strategy.Exercise 2. Apply the iterated elimination of strictly dominated strategies to thefollowing normal form games. Note that in some cases there may remain morethat one strategy for each player. Say exactly in what order you eliminated rowsand columns.Exercise 3. Apply the iterated elimination of dominated strategies to the followingnormal form games. Note that in some cases there may remain more that onestrategy for each player. Say exactly in what order you eliminated rows andcolumns.


Exercises

Exercises

Exercise 2 (cont.).


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