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A simple 4-player coalitional extensive-form game and its recursive backward induction solution - YouTube
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in this video I will solve a four-player
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correlational extensive form game by
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applying the recursive backward
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induction algorithm a coalitional
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extensive form game is an extensive form
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game with an addition of correlational
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utility function for each feasible
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coalition in this talk I assume that
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every coalition is feasible and that
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rational coalition's aim to maximize the
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minimum utility of their members in
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general coalition's may have any type of
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utility function as long as they are for
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Neumann Morgenstern utilities okay so
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coalition forming is defined as follows
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if some players I and J decide to form a
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coalition C so if C is equal to I and J
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then they each become an agent of this
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coalition of player C so I and J becomes
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an agent of coalition player C so their
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choices are guided by the utility
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function of player C which is given by
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UC okay
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obviously this is true only if they
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choose to form a coalition in general
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they are free to do so but in the
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recursive backward induction solution
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individual players join a coalition only
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if they have strict incentives to do so
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okay now let's move on to the example
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as I have just mentioned correlational
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utility function you see over here is
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defined as the minimum of UI is the
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minimum of UI where I is a member of
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correlation C okay so here is a four
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player game in coalitional extensive
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form game okay so here is a 4 player
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coalitional extensive for game in this
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context it means that players can
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cooperate on top of being able to choose
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none cooperatively and I will illustrate
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the recursive backward induction
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algorithm on this game in four steps
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step a shows the standard sub game
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perfect equilibrium solution when every
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player chooses none cooperatively ok so
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and the solution leads to the outcome 30
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40 30 and 0 given the sub game perfect
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equilibrium step B shows that both
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player 2 and player 3 in this sub game
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over here can benefit from cooperating
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by choosing F
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as a result receiving a payoff of 60 and
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40 for player 3
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in which case both of them are better
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off okay so let me just show you this
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payoff so player 2 is better off here
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compared to the egg for induction I'll
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come over there and player 3 is also
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better off here the notation 2 3
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signifies the cooperation between player
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2 and player 3 okay
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anticipating this collusion player 1
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would then choose MT which leads to the
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outcome 2000 hundred okay and this is
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the solution in step B now let's move on
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to step C step C shows that both player
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1 and player 2 cooperate by best
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responding the best responding to the
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solution in step B so let's focus over
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here notice that if player 1 and player
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2 cooperates player 1 chooses and 1
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player 2 chooses a then both player 1
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and player through to become strictly
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better off compared to the solution in
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step B so let me just point out these
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two pairs and therefore player 1 and
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player 2 as shown over here they have
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strict incentives to form a coalition
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and best respond with the choices given
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there
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so the solution in step C is then
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and clear one to cooperate and choose M
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1 and a which leads to the outcome 30 40
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30 and 0 okay
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however moving on to step D notice that
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player 1 can anticipate this possible
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cooperation between player 1 and 2
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therefore anticipates this cooperation
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and would actually offer to call offer
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player 1 to cooperate if player 1
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chooses M 2 player 4 can then choose a
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which leads to the outcome 6000 60 which
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is better for both player 1 and player 4
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right and notice that the outcome over
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here was disastrous for player 4 that's
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why therefore now has incentives to
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offer player 1 to form a coalition in
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that case in that case let me also
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signify here be mutually beneficial
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outcome for player 1 and therefore is
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6000 60 and that is the solution of step
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D and also solution of the whole game
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because there is no longer any coalition
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or non cooperative play that players can
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improve upon okay so this is called then
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the recursive backward induction
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solution in which player 1 and 4
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cooperate and in the off path player 2
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and 3 cooperate as I have explained in
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step a and this is the final solution ok
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notice that the only reason Claire for
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colludes with player 1
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in the last step is the credible threat
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of cooperation between player 1 and
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player 2 in step C that was that was a
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credible threat to player 4 and it leads
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to very bad outcome as I mentioned for
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player for to make a point let's assume
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that everything else being equal
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supposed that the outcome 2000 50 is
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replaced by the outcome 3500 8 5 okay so
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that's an off path outcome obviously and
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we made this change then player 1 would
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actually immediately choose M 2 in the
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beginning of the game okay would
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immediately choose M 2 in the beginning
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of the game non cooperatively and then
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player for would choose F and that will
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be the outcome of the K in this case
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and notice that Claire for would not be
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willing to cooperate with player 1
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because player 1 in the first place
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does not have any incentives to
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cooperate with player 2 and the reason
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is that here play one receives 30
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whereas over there player one receives
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35 so the coalition between player 1 and
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2 which is shown in step C would not be
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credible anymore
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therefore player 4 is not willing to
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cooperate okay the final outcome will be
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this ok
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this example shows that of path
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coalitional credible threats play a
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critical role in determining the recurse
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backward induction solution okay let me
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raise over here which the solution is
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illustrated over there
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