How Sequential Games Can Be Solved By Backward Induction Economics Essay
Game theory has been widely acknowledged as an of import tool in many Fieldss. Its attractive force is outstanding and its importance is explained by Robert Aumann and Oliver Hart in the undermentioned manner:
“ Game theory may be viewed as a kind of umbrella or ‘unified field ‘ theory for the rational side of societal scientific discipline… it develops methodological analysiss that apply in rule to all synergistic state of affairs ” ( Aumann and Hart, 1992 ) .
It is non hard to understand the enthusiasm towards theories of games developed from assorted game types and game-solution analyses. This essay will concentrate on a peculiar kind of games, viz. , consecutive game and the work outing method of backward initiation.
Consecutive games are those in which participants take bends traveling or do moves at different times. This means that participants traveling subsequently in the game have extra information about the class of others ‘ actions. This besides means that participants who move foremost can frequently act upon the game. Each participant makes determinations conditional on what other participants have done.
See a consecutive game where there is an incumbent ( Macrosoft ) and an entrant ( Microcorp ) . Macrosoft decides on a selling scheme for its new package game. It can take either a slick run or a simple run. Macrsoft faces possible competition with “ legal ringers ” of its game from Microcorp. It moves foremost and Microcorp observes its action. Regardless of what Macrosoft chooses, Microcorp so has two options: come ining the market, or remaining out of the market. The two houses ‘ final payments are displayed in table 1:
Table 1: The final payments for package game
Macrosoft ‘s ad run
Slick Simple
( – $ 250, $ 380 )
( $ 100, $ 400 )
( $ 0, $ 430 )
( $ 0, $ 800 )
Payoff ( in $ 1,000s )
Microcorp ‘s entry determination
Stay out
Enter
Figure 1: The game tree for package game
( $ 380,000, – $ 250,000 )
( $ 430,000, $ 0 )
( $ 400,000, $ 100,000 )
( $ 800,000, $ 0 )
Microcorp ‘s entry decisionIn order to set up the set of schemes for either house, it is of import to place clearly non merely the participants ‘ moves but besides the order in which these moves are chosen and the information available to participants when they make determinations. An effectual manner of forming this information is by utilizing a game tree. A game tree will picture a way of drama in add-on to the participants, actions, results and final payments. The game tree for the package game, therefore, appears as follow:
Payoffs:
( Macrosoft, Microcorp )
enter
B
simple
slipperiness
Macrosoft
Microcorp
enter
stay out
stay out
Microcorp
a
degree Celsiuss
Macrosoft has two schemes: take slipperiness or choose simple. Microcorp, nevertheless, has four schemes since there are two nodes to see, B and degree Celsius, and two possible actions at each node, enter or remain out. These schemes are:
Choose to come in regardless of which run Macrosoft chooses ( enter, enter ) .
Choose to come in if Macrosoft chooses slipperiness, otherwise choose to remain out ( enter, remain out ) .
Choose to remain out if Macrosoft chooses simple and frailty versa ( remain out, enter ) .
Choose to remain out in both instances when Macrosoft chooses slipperiness or simple ( remain out, remain out ) .
Table 2 shows the strategic signifier of the game:
Table 2: Strategic signifier of the package game
Macrosoft
slipperiness simple
( – $ 250, $ 380 )
( $ 100* , $ 400* )
( – $ 250, $ 380 )
( $ 0, $ 800* )
( $ 0* , $ 430* )
( $ 100* , $ 400 )
( $ 0* , $ 430 )
( $ 0, $ 800* )
( Payoffs in $ 1,000s )
( enter, enter )
Microcorp
( enter, remain out )
( remain out, enter )
( remain out, remain out )
There are two pure scheme Nash equilibria to this game which are { slick, ( remain out, enter ) } and { simple, ( enter, enter ) } . These are the optimum results of the game as no participant would wish to divert from his chosen scheme given the other ‘s pick. However, the inquiry is which of these equilibria is more sensible. The best result can be found through a process called backward initiation. This procedure assumes that participants act rationally at each node. This means that they will move in their ain best involvements. Knowing this, a participant working to work out a game tree can confidently take suboptimal actions to his challengers until merely the most likely way remains. By making this, an opposition ‘s possible moves from the initial node to the final payment can be depicted ; leting the participant to invent a scheme for each of those likely moves and finally finds the equilibrium. The package game can therefore be solved utilizing this method of logical thinking:
Figure 2: Game tree of the package game
( $ 380,000, – $ 250,000 )
( $ 430,000, $ 0 ) : A
( $ 400,000, $ 100,000 ) : Bacillus
( $ 800,000, $ 0 )
Payoffs:
( Macrosoft, Microcorp )
slipperiness
degree Celsiuss
enter
Microcorp
B
stay out
Macrosoft
enter
a
Microcorp
simple
stay out
At node B, come ining the market gives Microcorp a loss of $ 250,000, while remaining out gives it a zero-payoff. Therefore, Microcorp would rationally take to remain out. Similarly, the possibility that Microcorp will remain out at node degree Celsiuss can be eliminated since its final payment for enter is higher than that for stay out. Therefore, of the four schemes available to Microcorp, rearward concluding indicates that its lone optimum scheme is to take enter at node B and remain out node degree Celsiuss.
By sniping the non-optimal moves from Microcorp ‘s determination nodes, Macrosoft ‘s picks now look as follows:
Figure 3: The new game tree of package game
Payoffs:
( Marcrosoft, Microcorp )
( $ 430,000, $ 0 )
( $ 400,000, $ 100,000 )
simple
slipperiness
Macrocorp
Macrosoft ‘s optimum scheme is obvious-choosing slipperiness as this yields a final payment of $ 430,000 alternatively of $ 400,000 from following simple run. Therefore, by looking in front and taking its opposition ‘s entry determination into history Macrosoft can avoid doing a error of $ 30,000. Consequently, the scheme profile – { slipperiness, ( remain out, enter ) } is called the sub-game perfect equilibrium ( SPNE ) ; it is besides a Nash equilibrium ( NE ) of the game. Since backward initiation holds that participants will play their optimum action at each determination node, the ensuing schemes will therefore take to a NE. However, it is of import to observe that a NE is non ever a SPNE. In peculiar, the other NE of the package game – { simple, ( enter, enter ) } is non a SPNE. This is because it violates the regulations of backward initiation which assumes that Microcorp would ne’er take enter at node B.
On the other manus, Microcorp may desire to get at the NE – { simple, ( enter, enter ) } . Since Microcorp prefers outcome B of ( $ 400,000, $ 100,000 ) to outcome A ( $ 430,000, $ 0 ) ( figure 2 ) , but it can non acquire at that place unless Macrocorp adopts the simple scheme. Microcorp may, hence, endanger to ever take enter. If Macrosoft were to believe the menace, it would believe that it would gain merely $ 380,000 by taking slipperiness and $ 400,000 by taking simple. However, Microcorp ‘s menace to come in is non believable and Macrosoft knows that once it chooses slick, Microcorp will take remain out irrespective of its committedness as stay out is merely its best move at node B. In this instance, Macrosoft has the advantage by going the first mover and can therefore bring on its challenger to remain out of the market. While Microcorp suffers the disadvantages of a 2nd mover unless it could believably perpetrate to ever follow the scheme ( enter, enter ) .
Figure 3: Centipede game
I II I Payoffs to ( I, II )
( 8, 19 )
( 0, 0 ) ( -1, 10 ) ( 9, 9 )
Effective as it is, backward initiation has revealed some restrictions. One of these has been disclosed in the well-known centipede game. Figure 3 illustrates the game in which two participants alternate in taking between halting and go oning the game. If a participant stops the game, each will have a zero final payment at that point. But if a participant chooses to go on, he is fined ?1 while the other is rewarded with ?10. The game continues until one of the participants stops or both reach the concluding final payments of ?8 and ?19 severally.
Travel
Travel
Stop
Stop St
Stop
Travel
Backward initiation suggests that participant I should halt the game at the really first move and gets a nothing final payment. Suppose that the game has reached the concluding determination node where participant I makes the last move. At this point, participant I has to take between Stop and Go. The lone rational pick here is to halt and pocket ?9 instead than make up one’s minding to go on and having a less final payment of ?8. This means that at the old determination node, participant II will take to halt the game, taking into history that participant I, who is rational, responds by taking Stop on the following move. This in bend implies that participant I, at the first determination node, now efficaciously considers between Stop and having a zero final payment or Go and losing ?1 when participant II rationally stops the game at the wining node. Player I, hence, should halt the game instantly. This result is the alone SPNE. However, it would be better if participant I continues the game until he can acquire ?9 by halting at the penultimate node, or, as a 2nd best, until the concluding unit of ammunition where he gets ?8. The inquiry is that if participant I, in pattern, truly chooses to halt the game at the first determination node.
Experimental grounds by Kelvey and Palfrey ( 1992 ) and El-Gamul et Al. ( 1993 ) shows that the logic of backward logical thinking is rarely followed by decision-makers. In peculiar, in a four-legged centipede game experimented by Kelvey and Palfrey, merely 7 % of participants stopped the game at the really first move with a maximal final payment of $ 6.40 at its caput. When the final payment was increased to $ 25.60, 15 % chose Stop at the first determination node. Even at the concluding node, merely 69 % of participants ( in the high-payoff centipede ) and 85 % of participants ( in the low-payoff centipede ) chose to stop the game.
In decision, the principle of using backward initiation seems strong since it can assist contract the figure of possible Nash equilibria. By looking frontward and concluding rearward, each participant can foretell what other participants will make at subsequent phases of the game. Therefore, he can judge the effects of his possible moves, presuming that participants are rational, and hence ; decides on the optimum move. However, backward initiation exhibits some restrictions as discussed in the centipede game where the statement rests on the anticipation of behavior off the equilibrium way. This arguably leads to the challenge of reason premise of game theory which needs farther justification.
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