EconS 425 - Sequential Move Games

1 EconS 425 - Sequential move GamesEric DunawayWashington State OrganizationEric Dunaway (WSU) EconS 425 Industrial Organization1 / 57 IntroductionToday, we ll continue with our overview of game theory by looking atwhat happens when players take turns choosing their actions, ratherthan moving at the same are known as Sequential move move Games add another layer of strategy to the decisionmaking of all agents Dunaway (WSU) EconS 425 Industrial Organization2 / 57 Sequential move GamesAs stated before, Sequential move Games are simply where the orderof movement example, suppose we had two players, and player 1 was able tochoose their action before player 2 could choose 2 is able to observe the action taken by player 1, then , one player will have an advantage over the other player inthis case, but determining which player has that advantage dependson the game Dunaway (WSU) EconS 425 Industrial Organization3 / 57 Sequential move GamesLet s return to the prisoner s time, however, we will let player 1 decide whether to choosesilence or betray rst.

2 Then let player 2 observe player 1 s action andrespond to else about the game remains the model this game as a Sequential move game, we must make use ofthe extensive form of the game (as opposed to the normal form thatwe have already seen).This is represented by a series of decision trees with the outcomes andpayo s at the Dunaway (WSU) EconS 425 Industrial Organization4 / 57 Sequential move GamesSilenceBetraySilenceBetrayPlayer 1 Player 2-1-1-500-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization5 / 57 Sequential move GamesPlayer 1 Player 2 Player 2 SilenceBetraySilenceBetraySilenceBetray- 1-1-500-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization6 / 57 Sequential move GamesTo analyze a Sequential move game, we must make use of a techniqueknown as backward need to look at the actions that each player can make in order fromthe later actions until the earlier , we work backwards until we get to the top of the game we are able to determine the best responses for players, we cansubstitute them up the extensive form until we are left with one ll start with both of player 2 s possible actions, since they occur atthe end of the Dunaway (WSU) EconS 425 Industrial Organization7 / 57 Sequential move GamesPlayer 1 Player 2 Player 2 SilenceBetraySilenceBetraySilenceBetray- 1-1-500-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization8 / 57 Sequential move GamesPlayer 2 SilenceBetray-1-1-50 Eric Dunaway (WSU)

3 EconS 425 Industrial Organization9 / 57 Sequential move GamesSilenceBetrayPlayer 2-1-1-50 Eric Dunaway (WSU) EconS 425 Industrial Organization10 / 57 Sequential move GamesSilenceBetrayPlayer 20-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization11 / 57 Sequential move GamesSilenceBetrayPlayer 20-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization12 / 57 Sequential move GamesSilenceBetraySilenceBetraySilenceBe trayPlayer 1 Player 2 Player 2-1-1-500-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization13 / 57 Sequential move GamesNow that we have determined player 2 s best responses to everypossible action we can move up the extensive form to player 1 this is a game with perfect information (everyone knowseverything about everyone), player 1 knows how player 2 will react toall of their possible , player 1 will make their choice taking into consideration player2 s can show this decision making process for player 1 by simplysubstituting up player 2 s responses in the extensive is known as the reduced Dunaway (WSU) EconS 425 Industrial Organization14 / 57 Sequential move GamesSilenceBetraySilenceBetraySilenceBe trayPlayer 1 Player 2 Player 2-1-1-500-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization15 / 57 Sequential move GamesPlayer 1 SilenceBetray-50-3-3 Eric Dunaway (WSU)

4 EconS 425 Industrial Organization16 / 57 Sequential move GamesNow, player 1 simply chooses whichever of their actions yields thehighest payo , since player 2 s responses are already taken that is complete, we simply reassemble the extensive form ofthe game and can see all of the strategies for each Dunaway (WSU) EconS 425 Industrial Organization17 / 57 Sequential move GamesPlayer 1 SilenceBetray-50-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization18 / 57 Sequential move GamesPlayer 1 SilenceBetray-50-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization19 / 57 Sequential move GamesSilenceBetraySilenceBetraySilenceBe trayPlayer 1 Player 2 Player 2-1-1-500-5-3-3 Eric Dunaway (WSU) EconS 425 Industrial Organization20 / 57 Sequential move GamesAs we can see, in equilibrium, player 1 will choose to betray player 2,and then player 2 will respond by betraying player is the same outcome as in the simultaneous move game.

5 This willalways happen when a simultaneous move game only has a single I were being picky, I would say that the equilibrium strategy forplayer 1 is Betray, while the equilibrium strategy for player 2 that a strategy is a collection of all the actions a player 2 has two di erent actions in this game (one for each of player1 s possible choices), and a complete strategy must include all of them,even if they aren t on the equilibrium m not too picky though in this Dunaway (WSU) EconS 425 Industrial Organization21 / 57 Sequential move GamesWhat if we had a game with more than one Nash equilibrium, like in"The Battle of the Sexes?"Perhaps moving sequentially can help us determine which outcome wewill arrive s rst assume that the husband gets to make their choice rst,then the wife gets to observe the husbands choice and make her basically breaks the original premise of the game.

6 Most marriageproblems can be solved (or created) with a simple text message, by Dunaway (WSU) EconS 425 Industrial Organization22 / 57 Sequential move GamesFightOperaFightOperaHusbandWife1300 0031 Eric Dunaway (WSU) EconS 425 Industrial Organization23 / 57 Sequential move GamesHusbandWifeWifeFightOperaFightOpera FightOpera13000031 Eric Dunaway (WSU) EconS 425 Industrial Organization24 / 57 Sequential move GamesAgain, we use the backward induction technique in order to nd theequilibrium outcome for this the wife moves last, we ll look at their best responses to all ofthe husband s possible we ll look at what the husband s best choice is, taking the wife sresponses into save a few slides, I m just going to analyze the game as a whole,step by is usually quicker, Dunaway (WSU) EconS 425 Industrial Organization25 / 57 Sequential move GamesHusbandWifeWifeFightOperaFightOpera FightOpera13000031 Eric Dunaway (WSU) EconS 425 Industrial Organization26 / 57 Sequential move GamesFightOperaFightOperaFightOperaHusba ndWifeWife13000031 Eric Dunaway (WSU) EconS 425 Industrial Organization27 / 57 Sequential move GamesFightOperaFightOperaFightOperaHusba ndWifeWife13000031 Eric Dunaway (WSU) EconS 425 Industrial Organization28 / 57 Sequential move GamesNotice how the wife s best response always led to one of the twopossible Nash should make sense.

7 The husband and wife always got the highestpayo s when they attended the same the husband knows this, however, he can select his actionknowing that whatever he chooses, the wife will follow him naturally, he chooses his most preferred activity; the opera in if the wife moved rst?Eric Dunaway (WSU) EconS 425 Industrial Organization29 / 57 Sequential move GamesWifeHusbandHusbandFightOperaFightOp eraFightOpera13000031 Eric Dunaway (WSU) EconS 425 Industrial Organization30 / 57 Sequential move GamesFightOperaFightOperaFightOperaWifeH usbandHusband13000031 Eric Dunaway (WSU) EconS 425 Industrial Organization31 / 57 Sequential move GamesFightOperaFightOperaFightOperaWifeH usbandHusband13000031 Eric Dunaway (WSU) EconS 425 Industrial Organization32 / 57 Sequential move GamesNow we see the opposite the wife knew that the husband would follow her wherever shechose to go, she was able to choose the activity that gave her thehighest payo ; the boxing ght in this on which player was able to move rst, the Nashequilibrium we reached was di erent.

8 Each player selected the Nashequilibrium that yielded them the highest payo .We call this Nash equilibrium a subgame perfect Nash equilibrium inthis subgame perfect Nash equilibrium is simply a Nash equilibrium thatsurvives backward Dunaway (WSU) EconS 425 Industrial Organization33 / 57 Continuous Action SpacesLike our simultaneous move game counterpart, the majority ofexamples in this class use continuous action s look at our woolly mammoth hunter example time, hunter 1 gets to choose his e ort level before hunter , hunter 1 is able to set o for the hunt before hunter 2 isable to. By displaying his intended e ort level through huntingequipment, traps, etc, hunter 2 is left to respond to hunter 1 s e ortlevel the next Dunaway (WSU) EconS 425 Industrial Organization34 / 57 Continuous Action SpacesThe maximization problem for hunteriremains the same,maxeiei(1000 e1 e2) 100eiWe can solve this problem using backward induction, just like we didwith the earlier that we must start with the nal mover (hunter 2), andwork our way back up the tree until we reach the rst mover (hunter 1).

9 We want to nd a best response function for hunter 2, and substitutethat into earlier stages of our Dunaway (WSU) EconS 425 Industrial Organization35 / 57 Continuous Action Spacesmaxe2e2(1000 e1 e2) 100e2We ll nd that nothing changes for hunter 2. Taking a rst-ordercondition with respect toe2yields, Meat e2=1000 e1 2e2 100=0and solving this expression fore2gives us our best response functionfor any given e ort level of hunter 1,e2(e1)=450 e12 This should make sense. For hunter 2, he is simply reacting to hunter1 s e ort choice just like he was back in the simultaneous move has changed for Dunaway (WSU) EconS 425 Industrial Organization36 / 57 Continuous Action SpacesThis is where things start to when we were looking at the earlier Games that we wouldsend the result of the later stages of the game up the tree to the earlierstages.

10 Then the earlier player would pick their best choice taking thatinto can do that even without a formal "tree" to look 1 s maximization problem is,maxe1e1(1000 e1 e2) 100e1but remember that hunter 1 gets to move rst, and knows exactlyhow hunter 2 is going to react to his choice of e ort. Intuitively,hunter 1 knows that hunter 2 s e ort is a function of his own e ort,and he wants to factor that into his own maximization problem,maxe1e1(1000 e1 e2(e1)) 100e1 Eric Dunaway (WSU) EconS 425 Industrial Organization37 / 57 Continuous Action Spacesmaxe1e1(1000 e1 e2(e1)) 100e1e2(e1)=450 e12We can simply substitute in the best response function for hunter 2into hunter 1 s maximization problem. This is equivalent to passingup the result of hunter 1 s choice up the "tree,"maxe1e1 1000 e1 450 e12 100e1=maxe1e1 550 e12 100e1 Eric Dunaway (WSU) EconS 425 Industrial Organization38 / 57 Continuous Action Spacesmaxe1e1 550 e12 100e1 Taking a rst-order condition with respect toe1yields,550 e1 100=0e 1=450and plugging this value back into the best response function forhunter 2 gives us,e 2=450 e 12=225 Eric Dunaway (WSU) EconS 425 Industrial Organization39 / 57 Continuous Action Spacese 1=450e 2=225 Interestingly, hunter 1 (the rst mover) exerts twice as much e ort ashunter 2 (the last mover).