2. Static Games with Complete Information

2. Static Games with Complete Information
2.1 Simultaneous-move Game with Discrete Strategies Simultaneous-move Game (SMG) : Players choose strategies simultaneously without knowing what strategies others took. Ex) rock-paper-scissors Thus, a SMG is an imperfect information game. Perfect info. game : When each player knows the full history of the game up to that point at the time of choosing strategies. Or, when each information set is a singleton(한원소집합, a set with exactly one element).

2.1 Simultaneous-move Game with Discrete Strategies Perfect info. game: each information set is a singleton. *singleton: 한 원소 집합 : Information set Player 2 knows that Player 1 chose A 1, 2 Action C C (2, 3) (3, 2) (4, 0) (3, 1) 2 Reaching at this point, both players 1 and 2 are aware of the fact that player 1 chose A and player 2 chose D 1 A D E B *a move: a single action taken by a player at a node *a strategy: a complete plan of actions F

Imperfect info game: at least one info set is not singleton.
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies Imperfect info game: at least one info set is not singleton. : Information set At this node, player 2 is not sure if player 1 chose A or B: player 2 is not aware of the history of game  Imperfect info game 1, 2 C (2, 3) (3, 2) (4, 0) (3, 1) 2 But here at this final node, both players know that 1 chose A and 2 chose C. 1 A D E B F

2.1 Simultaneous-move Game with Discrete Strategies Even the timing of strategy choice is not the same, if players are isolated and do not know about the other’s choice, it is still a SMG. That is, the simultaneity of choice is irrelevant but recognition of other choice is relevant. Ex) When playing the game of rock-paper-scissors, players chooses strategy at different time without knowing what strategies others took, the game is still a SMG. To repeat, even though a game is sequentially (dynamically) proceeded, if the info is imperfect, it is classified as a static/simultaneous-move game. In this chapter, we will deal with games of imperfect info (as above) under symmetric info.

2.1 Simultaneous-move Game with Discrete Strategies Info. is symmetric? Asymmetric Info: When info on game (rules of the game, order of strategy choice, payoffs, uncertain variables and their impact on the outcome, history of players) is partially known only to some players. Or, the knowledge of game components are not common knowledge. That is, when the payoff function is not common knowledge. In this chapter, we assume symmetric info; we assume that everybody knows how payoffs are determined and what payoffs are. Payoff Function: The rule that maps inputs (players’ actions or strategies) into payoffs. Ex) Player 1’s Payoff = f1 (player 1’s strategy, player 2’s strategy) Payoff function = f (선수, 전략, 균형에 대한 정보)

2.1 Simultaneous-move Game with Discrete Strategies To sum up, in chapter 2, we will discuss the Static-Imperfect info-SMG under symmetric-complete information. SMG Imperfect info. But Imperfect infoSMG? NO. (SMG이면 항상 불완벽정보게임) (불완벽정보게임이면 SMG일수도 Seq.MG 일수도 있음) Ex) 가위-바위-보는 불완벽정보게임 Ex) 가위-바위-보를 동시선택으로도 할 수 있고, (상대방에게 감추고) (상대방이 뭘 내는지 모르므로) 순차적으로도 할 수 있음 Symmetric Info (Complete Info) Games Asymmetric Info (Incomplete Info) Bayesian Games Static or Simultaneous Move games Chapter 2 Imperfect Info but complete info game Chapter 4-1 Imperfect info but Incomplete info game Dynamic or Sequential Move games Chapter 3 Perfect or Imperfect info game but complete info game Chapter 4-2 Perfect or Imperfect info game but incomplete info game

Equilibrium concept for this kind of games: Nash Equilibrium
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies Equilibrium concept for this kind of games: Nash Equilibrium Who’s John Nash? Cryptology X+SQRT(X) Jennifer Connelly

2.1 Simultaneous-move Game with Discrete Strategies Still wonder who he is? Check his homepage (as of , still exists) at He worked on the universe and gravity other than logic and game theory... John Forbes Nash, Jr. 1928: Born ( ; Bluefield, West Virginia, USA) 1948: After undergraduate study at Carnegie Mellon Univ., Graduate study at Princeton, single sentence in the recommendation letter  ‘This man is a genius.’ 1950: ‘Non-cooperative Games’ published 1957: Tenured at MIT(29 yrs old) *I got mine at 46. 1958: gone mad (schizophrenia)  came back to Princeton and became ‘the Ghost of the Fine Hall’ 1970s: got better; confessed that he lived on the ultra-logical plane… 1994: Together with J. Harsanyi(Hungarian-American) and R. Selten(Germany), received the Nobel Prize in economics. He claims it was his "most trivial work"! 2015: killed in a car crash in a taxi

2.1 Simultaneous-move Game with Discrete Strategies Why the Nobel Prize? All the equilibrium concepts developed after Nash eq.(NE; 1950) were integrated into NE eventually. (Harsanyi and Selten contributed to that) That is, for 50 years, all the smart people have been locked in Nash’s mind and couldn’t escape from it, just like the Son-o-kong (Goku; Japanese) on Buddha's palm.

2.1 Simultaneous-move Game with Discrete Strategies What’s NE? Later. For now, just some examples of SM games. Introduction of new products in electronics or new models in car industry  develop/produce/market their products without knowing what model the competing company will produce. Election (we don’t know who’s voting for whom) Penalty kick : goalie vs. striker(kicker) To study these games, need to prepare basic terms (Next)

2.1 Simultaneous-move Game with Discrete Strategies 2.1.1 Discrete Strategy Strategy: a complete plan of action; combination of several alternatives Ex) A can do this or that. I can do that or this. If A do this, I do that. If A do that, I do this. Action: actually chosen/executed strategy Several actions imply sequentiality (순차성  dynamic games). In this case, strategy and move/actions can be differentiated. But in SM games, a strategy is chosen only once. Thus, strategy and action can’t be differentiated. Thus, in this chapter, we will not differentiate the two (p84, Games of Strategy).

Pure strategy is further classified into a discrete strategy.
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.1 Discrete Strategy What if a player randomly mixes more than two actions (or strategies) Ex) Tennis When strategies are chosen randomly, these strategies are called ‘mixed strategies.’ More details later in chapter 2. Before the ‘mixed’ strategies, ‘pure strategies’ will be discussed, where the chance to choose a specific action is 1. Pure strategy is further classified into a discrete strategy.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.1 Discrete Strategy Many games have limited number of strategies. Ex) 3 strategies of dribble, pass, shoot in basket ball game. In other cases, there may be infinite number of strategies. Ex) Price strategy: $100, $100.1, $ We will consider discrete strategies first, because they are easier to deal with. Game table (a.k.a. game matrix, payoff table, normal form, strategic form, or bi-matrix) is commonly used for SMGs.

Player 2 (Column Player; 열선수)
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.1 Discrete Strategy For a 2-person game(player 1 and 2), a game table looks like a spreadsheet. Ex) R-P-Sc game: Do we expect an equilibrium for the game? What’s equilibrium? Something that doesn’t change… 3-person R-P-Sc game? 3D. (Next) Player 2 (Column Player; 열선수) Rock Paper Scissors Player 1 (Row Player; 행선수) (0, 0) (-1, 1) (1, -1)

2.1 Simultaneous-move Game with Discrete Strategies 2.1.1 Discrete Strategy 3-person game in normal form

2.1 Simultaneous-move Game with Discrete Strategies 2.1.1 Discrete Strategy Two players (1 and 2), each player has 3 strategies of R, P or S; Depending on the strategies chosen, a player get the payoff of 1 if s/he wins, -1, if lose, and 0 if they draw. The left number of each payoff combination represents the payoff received by the row player (because strategies are presented in rows). The right one belongs to the column player (because strategies are presented in columns ). Ex) If 1 chooses R and 2 chooses S, 1 gets 1 and 2 gets -1. Sometimes only row player’s payoff is shown for zero sum games such as above. (next slide)  

Player 2 (Column Player)
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.1 Discrete Strategy For zero-sum games, only row player’s payoff is shown in the game table. Player 2 (Column Player) Rock Paper Scissors Player 1 (Row Player) -1 1

Column Player (C) Left Middle Right Row Player (R) Top 3, 1 2, 3 10, 2
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 1) Example of NE To immunize ourselves to real situation, take a generic game example as below; If Row chooses Low(click), Column chooses Middle(click), payoffs? 5(Row), 4(Column) ; This is the best response C can take when R plays Low.; What is R’s best response when C chooses Middle?  Low Column Player (C) Left Middle Right Row Player (R) Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 9, 7 ends

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 1) Example of NE (cont’d) Once ‘Low’ and ‘Middle’ are chosen, do R and/or C have reasons to make other choices? NO. Ex) Can they collude to choose ‘Bottom’ and ‘Right’ so that both of them increase the payoffs to (9, 7)? NO. If C promises to choose ‘Right’ and R believes that, R will choose ‘Low’ in stead of ‘Bottom’ because it increases the payoff to (12, 3). As a result, C will receive 3. Therefore, the collusion will collapse.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 1) Example of NE (cont’d) Thus, once ‘Low’ and ‘Middle’ are chosen, no one will deviate from this.  no change, stable  it is called an ‘equilibrium.’ This kind of equilibrium is called ‘Nash Equilibrium.’ More strictly, NE is defined as follows. NE: A combination of players’ strategies (not payoffs; There are two (4,5)s.) that the change of them will not improve the payoffs any longer. Can we expect to have NE all the time? No. Ex) R-P-S What if we already know the result of R-P-S game (the equilibrium)? No fun. Can we expect to have multiple equilibria in some other games? YES.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 1) Example of NE (cont’d) Then how do we find the NE? We can check all the payoffs(12 combinations above) through cell-by-cell inspection or enumeration. No room for error. But very tiresome. Also, for complicated games, it takes too long. Thus, we will look for a better way to find NE next. Note: Signal & Screening : In the above game, how would the column player increase the payoff of 4 to 6 (click)? Make the row player believe that column player will choose ‘Left.’ If R believes C will choose L, it is rational for R to choose ‘bottom,’ b/c that’s the maximum payoff R can get under that belief.  it is called ‘signaling.’ Ex of signaling: North Korea’s threat. Leaving no choice but one. Can C increase the payoff to 7? No. C can get 7 only when s/he chooses ‘Right’ and when C chooses Right, R will chooses ‘Low.’ Thus C will get 3, instead of 7.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 2) NE as a system of belief and choice NE assumes each player will choose the best response. The problem is, how do they ‘respond’ when choices are simultaneous? However, people are playing SMGs.  This implies the fact that each player has a system that can replace the observation on others’ actual choices. One of the systems is ‘experience’ and ‘prediction.’ If similar games with similar players are repeated, predicting other player’s reaction is generally possible. In this case, non-best choices will not be selected in the long-run.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 2) NE as a system of belief and choice (cont’d) Another system is the logical inference: belief, with which a player thinks in the shoes of other player(s). That is, one can try to come up with a logic that if I was the other player, I would do this… This will work if the other player is doing the same and I know this and the other knows that I know this and so on… and this thinking is repeated indefinitely. Then how to break this circular chain of logic becomes crucial. We can break this. (Next)

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 2) NE as a system of belief and choice (cont’d) NE is the best accomplishment so far in the logic system on thinking about thinking. belief(소신, 所信) : The notion on what other player(s) would choose in SMGs by observation or logic. Belief is different from ‘anticipation’ or ‘forecasting’  With a belief, others’ action occurs simultaneously, not in the future.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 2) NE as a system of belief and choice (cont’d) In this respect, NE is the configuration of the best strategies to others’ strategies. Thus, in this view of the belief system, NE can be defined as follows. NE is the set of strategies (not set of payoffs) of each player and it satisfies the following: (1) Each player has the correct(올바른) belief on other players’ strategies. (2) Given the own belief on other players’ strategies, NE is the combination of the best strategies of the players.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (우위; 優位) For some games, a strategy may be uniformly superior to other strategies. In this case, finding NE becomes simple. Ex) Prisoners’ Dilemma Two murder suspects are arrested on the murder scene and confined in two separate rooms and each is interrogated. The police has the firm belief that these two committed a murder. However, no one witnessed the actual murder and no murder weapon was found If the proofs are concrete, they can be prosecuted for murder. If not, their confession is needed. Therefore, they are told as the following.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) ‘If you confess your accomplice killed this man, you become just an accessory(從犯, 幇(방)助犯)… you were just in the wrong place at the wrong time. So you just need to serve 1 year. If your friend doesn’t confess, he will be put away for 25 years for first-degree murder. If both of you don’t confess, both of you will have to serve 3 years for kidnapping. If both of you confess that the other killed the man, you will serve 10 years each for 2nd-degree murder because you are cooperative to police. Don’t you think your accomplice confesses? The first to confess will be commuted(감형).’ This can be formulated into a SM 2-person game. (Next)

Prisoners’ Dilemma Game
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Payoff is the year to serve. Thus, the longer, the worse. Prisoners’ Dilemma Game Yoo, O-sung Confess Deny Jang, Dong-Gun 10, 10 1, 25 25, 1 3, 3

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Dong-gun (DG) should think about what strategy O-sung(OS) will take; If DG thinks OS will confess, DG will confess (10 years to serve rather than 25 years). Even when OS will deny, DG will confess because he will serve 1 year rather than 3. That is, DG will confess no matter what belief he has on OS’s choice. (click) Yoo, O-sung (OS) Confess Deny Jang, Dong-Gun (DG) 10, 10 1, 25 25, 1 3, 3

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) In this case, ‘confess’ is a dominant(우위) strategy for DG. If DG is a rational player, he has to confess. ‘Deny’ is called the ‘dominated(열위) strategy.’ To OS, ‘confess’ is the dominant strategy as well. Thus, both choosing ‘confess’ is expected in this game. This is also the NE of the game (‘Confess’ is the best response for both of them). Regardless of the belief on each other, ‘confess’ is the best strategy. This is the definition of ‘dominance.’

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) The Prisoners’ Dilemma game of this kind has 3 common aspects. 1) Each player has 2 strategies: Strategy 1. cooperate when other cooperates or defects. Strategy 2. defect when other cooperates or defects. 2) Each has a dominant strategy. 3) To both of them, the payoff from dominant strategy is less than the payoff from dominated strategy: lose-lose game. Why is this type of game important? 1) Many such games are found in economic, political and societal situations. 2) Both will lose after the game. Thus, studies have been done on how to avoid the dilemma and increase the payoff. More details later.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) What if only one player has a dominant strategy? If the other player has a dominant strategy, one can have the belief that the other will take the dominant strategy and choose the strategy. Next example. Game between the Congress(tax and expenditure policy) and Ministry of Finance(monetary policy, specially interest rate (IR) policy) The Congress favors low tax and high expenditure (Deficit). Why? Voters want such. But high chance of inflation. The Ministry of Finance(MF)’s primary task is to prevent inflation(lower value of money OR higher value of things). Thus, MF wants IR as high as possible.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) IR is the value(price) of money(price): S and D curves Congress wants to spend more money to please people. What happens to supply of money? IR Price Supply of Money Demand for Money Supply IR* Demand Quantity of Money Quantity of Things Increase in Money supply leads to Decreased IR (low IR or low value of money). When IR decreases, people prefer things over money  demand for things increases.  price for things increases  inflation.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) If IR is too low (value of money is too low), people want more things  things get expensive  inflation. Thus, MF wants IR high (to avoid inflation). But MF also faces political pressure from people to lower IR (loan demand for homes and automobiles people prefer lower IR). Thus, MF favors low IR but only so long as inflation is not a threat. The Congress: Deficit ≫ Balance, Low IR ≫ High IR MF: Balance ≫ Deficit (국회와 반대) 이자율(IR)은 경우에 따라 선호가 달라짐 1. Low IR ≫ High IR if Balance(국회가 예산을 알맞게 써서 인플레 없음), 2. High IR ≫ Low IR if Deficit(국회가 예산을 막써서 인플레).

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Thus, the Congress most prefers low IR ( cheaper price of money  lower tax effect; voters love it) and high expenditure(Deficit; voters are happy)  payoff=4. (국민도 좋아하고 국회도 좋아함) On the contrary, the Congress least favors high IR and low expenditure (Balance)  payoff=1. (국민도, 국회도 싫어함) Likewise, the Congress favors low IR and balance (싼 이자-인플레 위험 있지만 재정균형) next  payoff=3. Finally, the Congress favors High IR and Deficit (비싼 이자-인플레 위험 없지만 재정적자) next (payoff=2). Low IR+Deficit >> Low IR+Balance >> High IR+Deficit >>High IR+Balance >> >> >> Ministry of Finance Low IR High IR The Congress Budget Balance 3, ? 1, ? Budget Deficit 4, ? 2, ?

Ministry of Finance (MF)
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) MF most favors balance and low IR (when the chance of inflation is lowest)  payoff=4. Next is Balance-High IR combo. Payoff=3. MF least favors Deficit-Low IR combo because the chance of inflation is highest.  payoff=1. The next least favored combo is Deficit-High IR combo. Payoff=2. Every payoff is put together in the game table below. What are the dominant strategies? Ministry of Finance (MF) Low IR High IR The Congress (CG) Budget Balance 3, 4 1, 3 Budget Deficit 4, 1 2, 2 NO 인플레 인플레

2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) MF : If CG takes Balance, take Low IR. If CG takes Deficit, take High IR; No dominant strategy. CG : If MF takes Low IR, take Deficit. If MF takes High IR, take Deficit; Deficit is the dominant strategy. When MF knows this (has the belief that CG takes Deficit anyway), MF takes High IR (2 > 1). Thus, (Deficit, High IR) is the equilibrium. (click) Ministry of Finance (MF) Low IR High IR The Congress (CG) Budget Balance 3, 4 1, 3 Budget Deficit 4, 1 2, 2

2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) The problem is that there exist a combination of payoffs that is better for both of them  Balance-Low IR. Is this achievable? NO. Why? Let’s assume that CG and MF made a deal to take Balance (CG) and Low IR (MF). But if CG breaks the deal and take Deficit instead, MF will get the least favored payoff of 1. CG will get the most favored payoff of 4. (click) Ministry of Finance (MF) Low IR High IR The Congress (CG) Budget Balance 3, 4 1, 3 Budget Deficit 4, 1 2, 2

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Would CG break the deal? Of course… Never underestimate the flexibility of politicians’ mind. In reality of many countries, CG tends to execute the budget recklessly and MF utilizes High IR policy to prevent inflation. What problem do we have if IR is too high? The value of money increases people want to keep money  investments decreases  deflation. Recently in Korea and other developed countries, IR is very low. Is there a systematic way to use the concept of dominance to find an equilibrium when we have more than 2 strategies? YES. Next.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Successive or Iterated Elimination of Dominated Strategies (to find equilibria). So far we considered only 2 strategies mostly. Thus, one is the dominant strategy and the other is dominated strategy. Taking the dominant strategy is as same as removing dominated strategy. What if we have more than 2 strategies? A single strategy may not be dominant over all other strategies. However, once the dominated strategies are removed, a new game can be designed and solved.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) If, after removing the dominated strategies, only one outcome is left, these games are called ‘dominance solvable.’ The strategy combination for the outcome of the game becomes NE. Ex) Game in slide 18. R’s strategy ‘High’ is dominated by ‘Bottom’; 4<5, 3<4, 6<9  remove ‘High.’ (click)

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Original Game Once ‘High’ is removed, the game reduces to the table below. Now, C’s strategy ‘Left’ is dominated by ‘Right.’ (1<2, 2<3, 6<7) ; Thus, remove ‘Left.’ (click) In the original game, this was not possible; when R takes ‘High,’ C could get 5 for ‘Left’ and 4 for ‘Right.’ Thus, ‘Right’ did not dominate ‘Left.’ Thus, ‘Left’ could not be removed. That’s is, it(removing ‘Left’) became possible after ‘High’ has been removed. Column Player (C) Left Middle Right Row Player (R) Top 3, 1 2, 3 10, 2 Low 2, 2 5, 4 12, 3 Bottom 5, 6 4, 5 9, 7

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) After C’s ‘Left’ is removed, the game is reduced to the following. Now, R’s ‘Top’ and ‘Bottom’ are dominated by ‘Low.’ (2<5, 10< 12; 4<5, 9<12). Thus, ‘Top’ and ‘Bottom’ are removed. (click) Column Player (C) Middle Right Row Player (R) Top 2, 3 10, 2 Low 5, 4 12, 3 Bottom 4, 5 9, 7

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Now, only one strategy of ‘Low’ is left for R. Thus, C takes ‘Middle’ (4>3). (click) Payoffs are 5 for R and 4 for C. Thus, this game is ‘dominance solvable.’ When introducing this game, we asserted that this outcome is the NE of the game. We now understand how the NE is found. However, there are games that are not dominance solvable. Column Player (C) Middle Right Row Player (R) Low 5, 4 12, 3

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) If the game is not dominance solvable, is this method useless? NO. It still can reduce the size of the game so that we can solve the game more easily with other methods we will learn. So far, we considered the cases where payoffs from choice of strategies are bigger or smaller. But what if they tie(same payoffs)? Even when they tie, one of the tied strategies can be removed. In this case, it is said that ‘weakly dominated’ strategy is removed. Warning : Remove of weakly dominated strategy may remove the NE together. Next example.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) Ex of the problem. In the table below, ‘Top’ is weakly dominated by ‘Bottom’ for R (0<1, but 1=1)  Remove ‘Top.’ (click) ‘Left’ is weakly dominated by ‘Right’ for C as well  remove Left. (click) Thus, the equilibrium is Bottom-Right. However, Bottom-Left and Top-Right are NE as well. Why? Next. Column Player (C) Left Right Row Player (R) Top 0, 0 1, 1 Bottom

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 3) Dominance (cont’d) (Bottom, Left): If R takes Bottom, C can’t improve its payoff by taking Right. If C takes Left, R should stay on Bottom(no reason to change to Top)  NE (Top, Right): Check for yourself.  NE (Right, Bottom): Check for yourself.  NE Thus, if weak dominance is found, we need cell-by-cell check. Is there a better way?  YES. BR!! (NEXT) Column Player (C) Left Right Row Player (R) Top 0, 0 1, 1 Bottom

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 4) Best-Response (BR) Analysis In many games, there is no dominance. Also, several dominated strategies may be found but no single outcome of the game is found. Then, how can we find the expected outcome of the game (equilibrium)? We are still looking for NE but a simple elimination of dominated strategy is not enough. We need more sophisticated strategic thinking. From now on, we focus on new system of finding NE. For now, we do not consider the correctness of beliefs. But, we focus only on what is the best choice for the players.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 4) Best-Response Analysis (cont’d) That is, try to locate the best responses against all possible strategies of others. Look at the previous ex. From the view point of R, the best response to C’s ‘Left’ is ‘Bottom.’ Circle the payoff 5 (click). Likewise, circle the best payoffs when C takes ‘Middle’ and ‘Right.’ (click x 2) Column Player (C) Left Middle Right Row Player (R) Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 9, 7

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 4) Best-Response Analysis (cont’d) Next, make the circles from C’s position; When R takes ‘Top,’ C’s best response is to take ‘Middle’ and so on. Find a combination that has two circles (strategies are best responses to both)  Low-Middle This is the NE of the game. (check; start from High-Left  R changes to Bottom  C to Right  R to Low  C to Middle; R stays on Low  NE) Column Player (C) Left Middle Right Row Player (R) Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 9, 7

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 4) Best-Response Analysis (cont’d) This method of Best-Response Analysis find all possible NE in the game. Apply this method to the Prisoners’ Dilemma game. DG takes ‘Confess’ when OS takes ‘Confess.’ DG takes ‘Confess’ when OS takes ‘Deny.’ OS reacts the same. Thus, NE is (Confess, Confess) Yoo, O-sung (OS) Confess Deny Jang, Dong-Gun (DG) 10, 10 1, 25 25, 1 3, 3

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 4) Best-Response Analysis(BR) (cont’d) If NE is not found using BRA in the games of discrete strategy, there does not exist NE in the form of pure strategies in that discrete strategy game. In the strategy games with more strategies, the chance of finding NE increases. Why? There are almost infinite number of strategies as show below and therefore, the chance of having best responses for both of them(cross point) increases. (the idea of fixed-point theorem) g:f(x)-x a=0.2 b=0.8 less discrete strategies more discrete strategies continuous strategies Fixed Point Theorem [참고] Fixed Point Theorem(고정점 정리): 위 정사각형의 왼쪽 끝면(청색선)에서 오른쪽 끝면(적색선)을 연결하는 모든 연속적 선은 연두색 대각선을 지나가야 한다. *내쉬가 이를 이용해 연속전략하(예: 혼합전략)에서 NE가 항상 존재해야 함을 증명 Proof: Consider the function g: f(x) - x. g=f(a)-a ≥ 0 and g=f(b)-b ≤ 0. 중간값정리(intermediate value theorem)에 의하여, g 는 [a, b] 구간에서 ‘0’이 되는 점이 존재한다.; 이 ‘0’이 고정점이다. *중간값 정리 예) 일직선에서 2와 4는 중간값 3을 항상 갖는다.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 5) Mini-Max Method (for reference) Historically, Mini-Max method was developed before BRA. BRA is more powerful. We learn Mini-Max just because we need to know the history of game theory. Mini-Max is only applicable to zero-sum games. In 0-sum games, the most advantageous outcome to one player is the most disadvantageous to the other player, by definition. Or the other player will take the strategy that leads my worst outcome. Thus, I have to find the most advantageous payoff among worst scenarios (maximum among minima or maximin).

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 5) Mini-Max Method (for reference; cont’d) Ex) American football. Payoff is the advanced distance in yard (1 yard=0.914m) Offense’s payoffs : maximum is 10 when O takes ‘Long Pass’ and D reacts to Run. Defense : Take the strategy that minimize O’s payoff. Ex) If O takes Run, D takes ‘React to Run’  min=2. D’s payoff is suppressed. Blitz : 1) German for "lightning(번개)“, the German aerial attacks on Britain in WWII 2) Attack on the passer (quarterback) Defense (D) React to Run React to Pass Blitz Offense(O) Run 2 5 13 Short Pass 6 5.6 10.5 Medium Pass 4.5 1 Long Pass 10 4 -2 min=2 min=5.6 min=1 min=-2 max=10 max=5.6 max=13 *For longer pass, it takes time to run backwards. Run is a frontal breakthrough.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 5) Mini-Max Method (for reference; cont’d) Offense’s minima = {2, 5.6, 1, -2} Defense’s maxima = {10, 5.6, 13} Thus, O tries to take the maximin (maximum among minima= 5.6) and D tries to take minimax (minimum among maxima=5.6) Minima에서 max를 고르는 것은 max min Maxima에서 min을 고르는 것은 mini max Defense (D) React to Run React to Pass Blitz Offense (O) Run 2 5 13 Short Pass 6 5.6 10.5 Medium Pass 4.5 1 Long Pass 10 4 -2 O’s minima min=2 5.6 min=5.6 <=maximin min=1 min=-2 O’s Maxima max=10 max=5.6 max=13 =>minimax

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 5) Mini-Max Method (for reference; cont’d) If the maximin and the minimax coincide in one cell, the outcome is the NE (O’s short pass and D’s pass are best response to each other). In full description, it should be called the Maximin-minimax Method. However, it is called the Minimax Method for short. Defense (D) React to Run React to Pass Blitz Offense (O) Run 2 5 13 Short Pass 6 5.6 10.5 Medium Pass 4.5 1 Long Pass 10 4 -2 min=2 min=5.6=maximin min=1 min=-2 max=10 max=5.6 max=13 =minimax

Player 2 (Column Player)
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 5) Mini-Max Method (for reference; cont’d) What if the Minimax method can’t find NE? There is no NE for pure strategy in that (zero sum) game. (Ex below) Also, the Minimax method can be only applied to zero-sum games. Player 1’s minima Player 2 (Column Player) Rock Paper Scissors Player 1 (Row Player) -1 1 -1 Player 1’s Maxima

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 6) 3-Person Games Game description: 3 ladies (from the top; Jung, Da-Yeon; Lee, So-Ra; Lee, Young-Ae) decided to make a street garden. With more participating ladies, a better garden will be made and associated utility will be increased. However, in the case when the other two participate and I drop out, the garden will be less beautiful but I gain bigger utility because I don’t pay a dime. Payoff table for SMG can handle only upto 2 players? NO. It can handle more than three. Next.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 6) 3-Person Games (cont’d) Payoffs Number of Participating ladies Payoffs for Participating ladies Payoffs for Non-participating ladies 1 {a}, {b}, or {c}  1 평균적 정원, 참여자는 지불해야 함 {b,c}, {a,c}, or {a,b}  4 평균적 정원, 비참여자는 지불하지 않음 2 {a,b}, {a,c}, or {b,c}  3 평균이상 정원, 참여자는 지불 {c}, {b}, or {a}  6 평균이상 정원, 비참여자는 지불 안함 3 {a, b, c}  5 최고의 정원, 참여자는 지불 { }: no non-participating ladies 최고의 정원, 비참여자 없음, 보상 정의 안됨 { } : no participating ladies, no garden 정원 만들어지지 않음 {a}, {b}, and {c}  2 정원 없지만 누구도 지불하지 않음

So-Ra does not participate
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 6) 3-Person Games (cont’d; using dominant strategies) When So-Ra participates; For DY, ‘Don’t’ is the dominant strategy (If YA participates, 5<6; If YA doesn’t, 3<4) So-Ra participates Young-Ae Participate Do not particpate Da-Yeon 5, 5, 5 3, 6, 3 Don’t 6, 3, 3 4, 4, 1 So-Ra does not participate Young-Ae Participate Do not particpate Da-Yeon 3, 3, 6 1, 4, 4 Don’t 4, 1, 4 2, 2, 2 When So-Ra does not participate; For DY, ‘Don’t’ is the dominant strategy (If YA participates, 3<4; If YA doesn’t, 1<2) *For DY, ‘Don’t’ is the dominant strategy regardless of what SR or YA does

2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 6) 3-Person Games (cont’d; using dominant strategies) When So-Ra participates; For YA, ‘Don’t’ is dominant strategy (5<6; 3<4) So-Ra participates Young-Ae Participate Do not participate Da-Yeon 5, 5, 5 3, 6, 3 Don’t 6, 3, 3 4, 4, 1 So-Ra does not participate Young-Ae Participate Do not particpate Da-Yeon 3, 3, 6 1, 4, 4 Don’t 4, 1, 4 2, 2, 2 When So-Ra does not participate; For YA, ‘Don’t’ is dominant strategy (3<4; 1<2) *For YA, ‘Don’t’ is the dominant strategy regardless of what SR or DY does

2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 6) 3-Person Games (cont’d; using dominant strategies) When YA and DY participate, SR gets 5 if she participates, get 6 if she doesn’t. Compare payoffs of SR in the top table and the bottom table. ‘Don’t’ is the dominant strategy for SR as well. So-Ra participates Young-Ae Participate Do not participate Da-Yeon 5, 5, 5 3, 6, 3 Don’t 6, 3, 3 4, 4, 1 So-Ra does not participate Young-Ae Participate Do not particpate Da-Yeon 3, 3, 6 1, 4, 4 Don’t 4, 1, 4 2, 2, 2 Conclusion : The equilibrium is ‘nobody participates’ (the worst total payoffs among all possible payoffs). Another example of Prisoner’s Dilemma (next)

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 6) 3-Person Games (cont’d; using dominant strategies) If all participate, the total payoff is 15(=5+5+5). If nobody participates, the total payoff is 6(=2+2+2). Thus, in this case, collection of tax and making the garden by government is ideal. A public street garden is called the ‘positive externality.’ *Negative externality? Pollution. The same conclusion is reached by cell-by-cell inspection. By BRA? Much simpler. Next slides.

2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 6) 3-Person Games (using BRA) When So-Ra participates; So-Ra participates Young-Ae Participate Do not participate Da-Yeon 5, 5, 5 3, 6, 3 Don’t 6, 3, 3 4, 4, 1 When So-Ra does not participate; So-Ra does not participate Young-Ae Participate Do not participate Da-Yeon 3, 3, 6 1, 4, 4 Don’t 4, 1, 4 2, 2, 2 *Nobody participates is the NE with BRA as well.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games JP JD So far, most of the games have a unique pure strategy Nash equilibrium However, in general, a game needs not to have a unique equilibrium To consider this, think about a ‘coordination game(조정 게임).’ In a coordination game, players share some common interests but, because they act independently, they need to ‘coordinate’ their actions to gain mutually beneficial outcome. To think about this, take the example of Ku, June-Pyo (JP) and Keum, Jan-Dee (JD).

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) Meeting game : JD from the college of education (CE, 사범대)) at CBNU and JP from the CALES(농생대) at CBNU meet at the library and fell in love at the first sight. They wanted to chat forever but they had classes to go. When saying goodbyes, they promised to each other to meet again at 5:00. (Focal Point? Library) The problem was that they were under the heavy influence of the sex hormone and forgot to set the place to meet, whether at JD’s college (CE) or JP’s college(CALES). The worse is that they didn’t even ask each other’s name or phone numbers. What do you expect to happen? This game is expressed in the normal form (game table) next.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) If both choose the same place, payoffs are 1 to each. If they choose different places, payoff is 0. Using BRA, we can find 2 NEs (CE-CE, CALES-CALES). In either equilibrium, both will be happy (same payoffs). What is important for JD and JP is to choose the same place. Which action is irrelevant. Thus, this kind of game is called a “pure coordination game.” 사범대로 갈까? 농생대로 갈까? Jan-Dee CE CALES June-Pyo 1, 1 0, 0

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) Will the coordination be successful? JD could go to CALES for JP and JP could go to CE for JD. If JD knows this, JD might stay at CE and JP might wait for JD at CALES. If, again, the above fact is known to each, JD will try to go to CALES and JP will go to CE. Thus, there should be a device that can coordinate the beliefs or expectations on each other’s action. This is similar to the game of ‘flat tire’ in section 1.1. This coordination device is called the Focal Point (期待收斂點), in which the expectations converge.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) Examples of Focal Point What if JD’s leg was hurt and JP knew this? Both know the JD will have problem of walking and JD can expect that JP will come to CE and JP can expect that JD expects this. What if JD mentioned during the conversation that Joong-Moon(中門) is near to CE and there are many cafés to visit for talks? Like this, the chance of having the focal point depends on historical (to watch a big soccer match, cheering squads expect to gather in Kwang-Hwa-Moon) or cultural(butterflies fly to flower, not the other way around), and/or linguistic factors.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) JK JW We can modify the game as follows; They can meet at CE so that they can go to a café. But what if JD has a boyfriend, Jee-Who, who is a student at CE? What if JP’s girlfriend, Jay-Kyung, is also a CE student? If so, they might be better-off meeting at CALES b/c there is less chance for their affair to be exposed to their boyfriend or girlfriend.  the payoff when they meet at CALES is changed to (2, 2). Still two NEs. But the payoff when meeting at CALES is bigger. Jan-Dee CE CALES June-Pyo 1, 1 0, 0 2, 2

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) However, just a bigger payoff for meeting at CALES does not guarantee the certainty of meeting. This is because, for the probability of meeting at CALES to be bigger, JP needs to know that JD’s boyfriend is a CE student and JD needs to know that JP’s girlfriend is a CE student. Also, they need to know that the other prefers to meet at CALES as well. Also, the fact that they know this should be known to the other… That is, the payoffs should be a common knowledge (常識; 주지의 사실); Both of them need to know all the payoffs in all possible cases exactly.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) For this, they have had talked about the advantages of meeting at CALES and agreed but did not finalize the meeting place.  Without a certain focal point there is a high chance of finding another equilibrium. In this sense, the above game and such is called the ‘assurance game(확신 게임).’ In reality, this assurance is easily secured. For example, if one said ‘I will go to CALES,’ the other one has no reason to disbelieve what is said. Then, the best payoffs will be acquired. If there is a conflict of interest, the sincerity of information exchange can be more problematic. We will deal with this in asymmetric info. game.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) If a group for the game is large, they can publicize the information exchange (public announcement). The problem may be whether or not the players are paying attention to this announcements. Now, the game is altered again; What if JP’s girlfriend (JayKyung) is a CALES student and JD’s boyfriend (JeeWho) is a CE student? JP prefers to meet at CE and JD prefers CALES. That is, what if preferred strategies are different?  conflicts of interests; called the ‘battle of sexes(성대결 게임).’ Jan-Dee CE CALES June-Pyo 2, 1 0, 0 1, 2

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) The name of the game is titled when the sexual discrimination prevailed in 50’s. In this game, a couple try to coordinate if they go to a boxing match or the ballet. Why is this a sexist game? It assumes that men love a boxing match and women love the ballet… In these days, women love K1 and the boxers too.

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) However, the battle of sexes game does not always imply sexual discrimination. Then, what outcome do we expect? Multiple equilibria again. But we have higher chance of unsuccessful coordination. Why? Asymmetric payoffs; preferences differ. High chance of ruthless actions to make the outcome more advantageous to oneself. Or, on the contrary, they might act altruistically(利他的) and JP goes to CALES for JD (when JP’s girlfriend is JK, a CALES student). If this is true, JP’s altruism should be reflected in the payoff table  Next slide Jan-Dee CE CALES June-Pyo 2, 1 0, 0 1, 2 Ruthless Altruistic

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) Altruistic Case: Both altruistic : JP는 재경에게 걸릴 것을 무릅쓰고 CALES에 남고, JD는 지후에게 걸릴 것을 무릅쓰고 CE에 남는다 Can’t meet Jan-Dee CE CALES June-Pyo 2, 4 3, 3 4, 2 - JP, JD 모두, 만나는 것 좋지만 상대방이 안 걸리는 것 선호(4) - 다음은 못 만나도 상대방이 안 걸리는 것 선호(3) … Ruthless Case: Both ruthless  JP는 재경에게 걸릴 것을 피하려고 CE로 가고, JD는 지후에게 걸릴 것을 피하려고 CALES로 간다  Can’t meet Jan-Dee CE CALES June-Pyo 4, 2 3, 3 2, 4 JP, JD 모두, 만나는 것 좋지만 자신이 안 걸리는 것 선호(4) 다음은 못 만나도 내가 안 걸리는 것 선호(3) …

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) Last modification: The game of chicken(닭게임). (이유없는 반항, Rebel without a Cause, 1955) *왜 index finger(가리키는 손가락, 집게 손가락)라고 부를까?

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 7) Multiple Equilibria in Pure Strategy Games (cont’d) The game is presented below. Two NEs. To win, one needs to be charismatic or to show that s/he is not a chicken. (by taking visible and irreverasable strategies; tying the steering wheel…) One more thing: What if a player takes a strategy by chance or randomly? NE? More details on this in the mixed strategy. 4-3-19 DEAN Swerve Straight JAMES 0, 0 -1, 1 1, -1 -2, -2 핸들 빼버리는 것이 가장 확실

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 8) No Equilibrium in Pure Strategy Game So far, we thought about the games where there is a unique equilibrium or there are multiple equilibria. However, some games have no equilibrium at all. This is the case players have no consistent strategies to take. Tennis match in Chapter 1; Navratilova vs. Evert. Down-the-line (DL) Evert (right handed) Navratilova (left handed) Cross-court(CC)

Navratilova(Defense)
2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 8) No Equilibrium in Pure Strategy Game (cont’d) When Evert takes DL and Nav takes DL, the chance to succeed for Evert is 50%. When Evert takes DL and Nav takes CC, the chance to succeed for Evert is 80%. (Nav’s success chance is 20%). Similarly, if Evert takes CC, the chance is 90% if Nav takes DL, 20% if Nav takes CC. Zero-sum (constant-sum, more generally) game. Dominance, minimax, or cell-by-cell can be used to find if there is any NE. Navratilova(Defense) DL CC Evert (Offense) 50% 80% 90% 20% *CC-CC 20%, DL-DL 50% Why? Nav. left-handed.

2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 8) No Equilibrium in Pure Strategy Game (cont’d) First of all, there is no dominant strategy. That is, if Nav takes DL, Evert should take CC (50%<90%) and if Nav takes CC, Evert needs to take DL (20%<80%). Same for Nav. If Evert takes DL, Nav should takes DL(100-80=20<100-50=50) and if Evert takes CC, Nav should take CC(100-90=10<100-20=80). Navratilova(Defense) DL CC Evert (Offense) 50, 50 80, 20 90, 10 20, 80

2. Static Games with Complete Information 2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 8) No Equilibrium in Pure Strategy Game (cont’d) Via cell by cell check: Start with DL-DL; If Evert takes CC, the chance increases to 90%. But if Nav takes CC, it decreases to 20%. If Nav takes CC, Evert takes DL and increases the chance to 80%. However, again, if Evert takes DL, Nav will take DL  circular change of strategies; players always have reason to change strategies  No (Nash) Equilibrium. Navratilova(Defense) DL CC Evert (Offense) 50% 80% 90% 20%

2.1 Simultaneous-move Game with Discrete Strategies 2.1.2 Nash Equilibrium (NE) 8) No Equilibrium in Pure Strategy Game (cont’d) Important point in these kind of games: What matters is not what other players DO. What matters is what other players DO NOT. Not repeating the same strategy is important.  act unsystematically so that others can’t anticipate your moves  This strategy (mixing strategy) is called ‘mixed strategy.’ *’가위-바위-보’에서 우리가 전략을 무작위로 섞는 전략을 쓰는 경우와 같음. When the ‘mixed strategy’(not pure strategy) is used, there always exists an equilibrium. 고정점정리로 증명 가능. More details later. Now we move onto continuous strategy games.

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