The Influence of Nature on Outcomes of Three Players Game

Transcription

1 The Influence of Nature on Outcomes of Three Players Game Dr. Sea-shon Chen, Department of Busness Admnstraton, Dahan Insttute of Technology, Tawan ABSTRACT Ths paper based on Nash equlbrum theory to explore the uncertanty of Nature that nfluences the outcome of the three player s game. Nature has two phases; one s exogenous uncertanty or random and another s endogenous uncertanty or strategy choosng. Exogenous random s the envronment or stuaton unpredctable. Endogenous uncertanty s the players unpredctable ntentons that result n mxed strategy n the game. The paper expresses the methodology and utlze computer to explore dscrete equlbrum of the game, then connect dscrete equlbrums nto contnuous results. The result reveals that pure equlbrums and mxed equlbrums exst n dfferent uncertanty range. The results also suggest, n the real world, the players know the nformaton of Nature s mportant because that wll nfluence decson makng and payoffs. Keywords: Nature, exogenous uncertanty, endogenous uncertanty, Nash equlbrum INTRODUCTION Game theory s a modelng tool. It s an nterdscplnary and dstnct approach to the study of human behavor and strategc management (Perea, et al., 2006; Rasmusen, 1995; Saloner, 1991). The dscplnes most nvolved n game theory are mathematcs, economcs, socal scence, and behavoral scence. The essental elements of a game are (1) players, (2) actons, (3) nformaton, (4) strateges, (5) payoffs, (6) outcomes, and (7) equlbrums. Players are the ndvduals who make decson based on that they are absolutely ratonal n ther economc choces. Game theory s based on the assumpton that human bengs are absolutely ratonal n ther economc choces (von Neumann and Morgenstern, 2004). Specfcally, the assumpton s that each person tres to maxmze her or hs rewards (utltes, profts, ncomes, or subjectve benefts) n the crcumstances that a player faces (Chen, 2008). Ths hypothess serves a double purpose n the study of the allocaton of resources. Frst, t narrows the range of possbltes; somewhat absolutely ratonal behavor s more predctable than rratonal behavor. Second, t provdes a crteron for evaluaton of the effcency of an economc system. Sometmes the rratonal pseudo-player, Nature, takes random actons at specfed ponts n the game wth specfed probabltes (Rasmusen, 1995). Nature s ndfferent to the outcomes, but the strategc players care about the outcomes. Although the strategc players choose the strateges and the choces affect the outcomes, the state of the world chosen randomly by Nature who s ether exogenous or endogenous uncertan probablty and ths nformaton revealng bascally affects the outcomes (Antono, 2006; Berman and Fernandez, 1998; Shmaya, 2006). The value of nformaton about Nature s very mportant; players may or may not know the probablty Nature wll choose (Ponssard, 1976). In many cases, the result we predct mmensely nfluences our decson. The msplay of decson s not due to shortage and/or error of nformaton or the logc reasonng, but the phase we thnk s unque (Cruz & Smaan, 2000; Zhu, 2004). We always put our hands to the phase that we customzed and overlook the changeful and uncertan realty. Eventually, our conservatve and sealed vson results n

2 faulty judgment and decson-makng. One of the game theory players, Nature, does not commt the msplay of decson. Nature plays the game n many phases. Uncertanty or random s the realty of the player who s Nature. Exogenous uncertanty s that the strategc game players do not know the rratonal pseudo-player has what knd of the state of the world. For example, three frms are decdng whether and how to drll wells nto a sprng water depost that les under ther adjacent land tracts. The three frms do not know for sure whether there s water under ther land or not,.e. uncertanty of the state of the world or the phase of player Nature. Endogenous uncertanty s that the strategc game players choose among ther acton randomly, but sometmes wth reasonng. In the non-cooperatve game, when more than one player adopts a mxed strategy (a probablty stated strategy) these players randomze ndependently of each other (Prasad, 2003; Ravkumar, 1987). Independence means that knowledge of the strategy chosen by one player provdes no new nformaton about the strategy that wll be chosen by any other player who has adopted a mxed strategy n contnuous or dscontnuous game (Berman and Fernandez, 1998; Reny, 1999). For example, TAI-water, -water, and HIB-water frm may choose don t drll, a narrow well, or a wde well on certan probablty. The probabltes whch the frms make decsons are endogenous uncertanty or psychologcal Nature. Some papers or books ntroduce or nvestgate exogenous and endogenous uncertanty n statc games whch lmted n a case or study two games separately. Ths paper studes two knds of Nature, exogenous and endogenous uncertanty, mpact on the equlbrum of games wth computer. By usng computer to study, the processes are easy and the results can be contnuous and llustrated. PRINCIPLE Exogenous and endogenous uncertanty n statc games may result n mxed strategy. In game theory a mxed strategy s a strategy whch chooses randomly among possble moves. The strategy has some probablty dstrbuton whch corresponds to how frequently each move s chosen. A totally mxed strategy s a mxed strategy n whch the player assgns strctly postve probablty to every pure strategy. A mxed strategy should be understood n contrast to a pure strategy where a player plays a sngle strategy wth probablty. Pure strategy Nash equlbrums are Nash equlbrums where all players are playng pure strateges. Mxed strategy Nash equlbrums are equlbrums where at least one player s playng a mxed strategy. Durng the 1980s, the concept of mxed strateges came under heavy fre for beng ntutvely problematc. Randomzaton, central n mxed strateges, lacks behavoral support (Shachat, and Swarthout, 2004). Seldom do people make ther choces followng a lottery. Ths behavoral problem s compounded by the cogntve dffculty that people are unable to generate random outcomes wthout the ad of a random or pseudo-random generator (Aumann, 1985). However, ratonal strateges exst for fnte normal form games under the assumpton that strategy choces can be descrbed as choces among lotteres where players have securty- and potental level preferences over lotteres (Zmper, 2007). Game theorst Rubnsten (1991) ponts out two alternatve ways of understandng the concept: one s to magne that the game players stand for a large populaton of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fracton of agents choosng each strategy. The mxed strategy hence represents the dstrbuton of pure strateges chosen by each populaton. However, ths does not provde any justfcaton for the case when players are ndvdual agents. The other, called purfcaton s to suppose that the mxed strateges nterpretaton merely reflects our lack of knowledge of the agent's nformaton and decson-makng process. Apparently random choces are then seen as

3 consequences of non-specfed, payoff-rrelevant exogenous factors. However, t s unsatsfyng to have results that hang on unspecfed factors, and ths dsmsses the possblty of a mxed-strateges analyss to have any predctve power. Argung that those factors are smply other players' belefs about a player's strategy, hence adoptng a mxed strategy s the best response to a player playng mxed strateges, gves a credble nterpretaton, but does not restore predctve power to the concept of mxed equlbrums. Although economst s atttude towards mxed strateges-based results has been ambvalent, mxed strateges are stll wdely used for ther capacty to provde Nash equlbrum n any game concernng mnmum monetary regret or maxmum profts (Bade, 2005; Chong and Benl, 2005). The followng s an example. Let (S, f) be a game, where S s the strategy set for player, S = S 1 S 2 S n s the set of strategy profles and f = (f 1 (x),..., f n (x)) s the payoff functon. Let x be a strategy profle of all players except for player. When each player belongs to {1,..., n} chooses strategy x resultng n strategy profle x = (x 1,..., x n ), then player obtans payoff f (x). The payoff depends on the strategy profle chosen,.e. on the strategy chosen by player as well as the strateges chosen by all the other players. A strategy profle x* belongs to S s a Nash equlbrum f no unlateral devaton n strategy by any sngle player s proftable for that player, that s * * * *, x S, x x : f ( x, x ) f ( x, x ). (1) In order to nvestgate mxed strateges, notng p s the payoff converted probablty and p = (p 1 (x),..., p n (x)) s the functon of the payoff converted probablty. A strategy profle x* belongs to S s a mxed equlbrum f no unlateral devaton n strategy by any sngle player has the most probable of the payoff converted probablty for that player, that s * * * *, x S, x x : p ( x, x ) p ( x, x ). (2) A game can have a pure strategy Nash equlbrum (NE) or a mxed NE n ts mxed extenson that of choosng a pure strategy stochastcally wth a fxed frequency. Nash explaned that, f we allow mxed strateges,.e. players choose strateges randomly accordng to the most probable probabltes then every n-player game n whch every player can choose from fntely many strateges admts at least one Nash equlbrum. METHODOLOGY A good model n game theory has to be realstc n the sense that t provdes the percepton of real lfe socal phenomena (Rubnsten, 1991). The paper based on ths argument and the prncples to nvestgate pure strategy and possble mxed strategy. A stuaton s three frms TAI,, and HIB are decdng whether and how to drll wells nto a mneral water resources that le under ther adjacent tracts. Because the three frms do not know for sure whether there s mneral water under ther tracts or not, the consequence of ther actons depends on Nature (exogenous uncertanty) that beyond ther knowledge and control. That s the outcome of the game depends on the state of the world chosen randomly by Nature (the state of the world) and the strateges chosen by the strategc players (TAI,, and HIB). Suppose there possble states of the world: ether there s a depost of x bllon gallon water under the land and any well wll be a gusher (probablty p), or there s no water under the land and any well wll be a dry hole (probablty q or 1 p). The two states of the world (Nature) and the possble strategy profles result n 54 possble outcomes. A three dmensonal matrx (3 3 3) s suggested to represent the payoff matrx of the mneral water drllng game when Nature s gushng (probablty p). For example, M 132 = ( p T 132,

4 p C 132, p H 132 ) means the proft for each frm f TAI chooses don t drll, chooses wde, and HIB chooses narrow; M 213 = ( p T 213, p C 213, p H 213 ) means the proft that TAI chooses narrow, chooses don t drll, and HIB chooses wde. Another matrx (3 3 3) represents the payoff matrx of the game when Nature s dry well (probablty q). For example, N 131 = ( q T 131, q C 131, q H 131 ) means the proft for each frm f TAI chooses don t drll, chooses wde, and HIB chooses don t drll; N 212 = ( q T 212, q C 212, q H 212 ) means the proft that TAI chooses narrow, chooses don t drll, and HIB chooses narrow. The expected payoff matrx G jk for the mneral water drllng game s suggested. For example, n ((3 3 3) expected payoff matrx G 312 = ( T u 312, C u 312, H u 312 ), the elements wll be counted as T u 312 = p T 312 p + q T 312 q, C u 312 = p C 312 p + q C 312 q, and H u 312 = p H 312 p + q H 312 q. Pure Strategy By usng Excel, f nput data nto matrxes M jk and N jk the resultng G jk wll provde nformaton for equlbrum judgment or advanced mxed strategy measurement. An emprcal case study wth dgts wll perform n the followng. Table 1 and Table 2 are the data of payoff matrx of the mneral water drllng game f gashng and f dry well, respectvely. Table 3 s the format of expected payoff matrx of the mneral water drllng game. Table 1: The Payoff Matrx of the Mneral Water Drllng Game f Gushng Nature: Gushng (probablty p) HIB (Don t drll) Don t drll TAI Narrow Wde Don t drll TAI Narrow Wde HIB (Wde) Don t drll TAI Narrow Wde Mxed Strategy If there s no pure strategy equlbrum, mxed strategy equlbrum wll be fgured out by the expected payoff and probabltes. Let p TD, p TN, and p TW denoted the probablty of TAI choosng Don t drll, Narrow, and Wde. Let p CD, p CN, and p CW denoted the probablty of choosng Don t drll, Narrow, and Wde. And, let p HD, p HN, and p HW denoted the probablty of HIB choosng Don t drll, Narrow, and Wde, respectvely. The mxed strategy profle wll be fgured out by comparng these probabltes. The process to fnd the most possble probablty set s frst computng the average and standard devaton of all the elements to get two 3 3 matrxes; one s the average and the other s the standard

5 devaton. For example, referrng to Table 3, the average of T u 1jk (j and k = 1, 2, 3) s the element of average matrx a 11 and the standard devaton of H u 2jk (j and k = 1, 2, 3) s the element of standard devaton s 23. Second, the standard normal random varable, Z, s calculated. Thrd, the probablty of each Z s calculated. Fnally, comparng the frm probablty, the combnaton of probablty for mxed strategy wll be found. Table 2: The Payoff Matrx of the Mneral Water Drllng Game f Dry Well Nature: Dry well (probablty 1 p) HIB (Don t drll) Don t drll TAI Narrow Wde Don t drll TAI Narrow Wde HIB (Wde) Don t drll TAI Narrow Wde Table 3: The Expected Payoff Matrx of the Mneral Water Drllng Game HIB (Don t drll) Don t drll T u 111 C u 111 H u 111 T u 121 C u 121 H u 121 T u 131 C u 131 TAI Narrow T u 211 C u 211 H u 211 T u 221 C u 221 H u 221 T u 231 C u 231 Wde T u 311 C u 311 H u 311 T u 321 C u 321 H u 321 T u 331 C u 331 H u 131 H u 231 H u 331 Don t drll TAI Narrow Wde T u 112 T u 212 T u 312 C u 112 H u 112 C u 212 H u 212 C u 312 H u 312 T u 122 C u 122 T u 222 C u 222 T u 322 C u 322 H u 122 T u 132 H u 222 T u 232 H u 322 T u 332 C u 132 C u 232 C u 332 H u 132 H u 232 H u 332 HIB (Wde) Don t drll TAI Narrow Wde T u 113 T u 213 T u 313 C u 113 H u 113 C u 213 H u 213 C u 313 H u 313 T u 123 C u 123 T u 223 C u 223 T u 323 C u 323 H u 123 T u 133 H u 223 T u 233 H u 323 T u 333 C u 133 C u 233 C u 333 H u 133 H u 233 H u 333 RESULTS Based on the prncple and methodology, the study assumes asymmetrcal date matrx (Table 1 and Table 2) and uses computer Excel to explore the pure strategy equlbrum and mxed strategy

6 equlbrums. The results are also explaned by three tables (Tables 4, 5, and 6) to express the pure and mxed equlbrum. Pure Strateges An example of asymmetrcal data matrxes s suggested. After analyss the data wth exogenous uncertanty (gushng probablty p = 0.90, dry well probablty q = 0.10) and the expected payoff matrx are lsted n Table 4. The unque strategy for Nash (pure) equlbrum of the game s {Wde, Wde, Narrow} because of T u 332 (1.5) > T u 132 (0) > T u 232 (-6), C u 333 (3.5) > C u 313 (0) > C u 323 (-5), and H u 332 (9) > H u 333 (1.2) > H u 331 (-1). Table 4: The Expected Payoff Matrx of the Mneral Water Drllng Game (p = 0.90) HIB (Don t drll) Don t drll TAI Narrow Wde Don t drll TAI Narrow Wde HIB (Wde) Don t drll TAI Narrow Wde If the range of gushng probablty, p, s 1.0 p , The expected payoff functons for TAI-water (u T ), -water (u C ), and HIB-water (u H ) are Equaton 3, 4, and 5. TAI: u T = p , (3) : u C = p , (4) and HIB: u H = p (5) Usng the former stated rule, for the range of p s 0.75 p 0.50, the unque strategy for Nash (pure) equlbrum of the game s {Narrow, Narrow, Narrow}, and the expected payoff functons are Equaton 6, 7, and 8. TAI: u T = 30 p 15, (6) : u C = 30 p 10, (7) and HIB: u H = 40 p 10. (8) For the range of p s 0.49 p , the unque strategy for Nash (pure) equlbrum of the game s {Don t drll, Wde, Wde}, and the expected payoff functons are Equaton 9, 10, and 11. TAI: u T = 0, (9) : u C = p , (10) and HIB: u H = p (11) For the range of p s p , the unque strategy for Nash (pure) equlbrum of the game s {Don t drll, Narrow, Don t drll}, and the expected payoff functons are Equaton 12, 13, and 14.

7 TAI: u T = 0, (12) : u C = p , (13) and HIB: u H = 0. (14) For the range of p s p 0.001, the unque strategy for Nash (pure) equlbrum of the game s {Don t drll, Don t drll, Narrow}, and the expected payoff functons are Equaton 15, 16, and 17. If gushng probablty, p = 0.00, the equlbrum strategy s {Don t drll, Don t drll, Don t drll}. TAI: u T = 0, (15) : u C = 0, (16) and HIB: u H = p (17) Mxed Strategy If the gushng probablty p s p 0.751, then the pure strategy does not exst. For example, f the stuaton s gushng probablty p = 0.80 and dry well probablty q = 0.20, the expected probablty matrx are lsted n Table 5. Table 5: The Expected Probablty Matrx of the Mneral Water Drllng Game (p = 0.80) HIB (Don t drll) Don t drll TAI Narrow Wde Don t drll TAI Narrow Wde HIB (Wde) Don t drll TAI Narrow Wde Under the condton of gushng probablty p = 0.8, the strategy for mxed equlbrum of the game s {Wde, Don t drll, Wde}, because TAI (Wde: p TW = 0.29), (Don t drll: p CD = 0.11), and HIB (Wde: p HW = 0.32) are the most possble combnaton of the probablty matrx. If the gushng probablty s p 0.751, the mxed strategy s also {Wde, Don t drll, Wde}, and the expected probablty functons for TAI-water (p T ), -water (p C ), and HIB-water (p H ) are Equaton 18, 19, and 20. TAI: p T = p p , (18) : p C = , (19) and HIB: p H = p p (20) If gushng probablty s p 0.230, the pure strategy does not exst. The mxed strategy s {Don t drll, Narrow, Narrow}. For example, f p = 0.3, the expected probablty matrx are lsted n Table 6.

8 Table 6: The Expected Probablty Matrx of the Mneral Water Drllng Game (p = 0.30) HIB (Don t drll) Don t drll TAI Narrow Wde Don t drll TAI Narrow Wde HIB (Wde) Don t drll TAI Narrow Wde Table 6 showed that TAI (Don t drll: p TD = 0.17), (Narrow: p CN = 0.10), and HIB (Narrow: p HN = 0.07) are the most probable stuaton combnaton for the gushng probablty p = Actually, n the range (0.333 p 0.230), the expected probablty functons are Equaton 21, 22, and 23. TAI: p T = 0.50, (21) : p C = p p , (22) and HIB: p H = p p (23) DISCUSSION AND CONCLUSION The results reveal that Nature nfluences the outcome of the games and makes the results to be very complex especally f the players are three or more than three (Daskalaks and Papadmtrou, 2005). In the results, the pure equlbrum strategy profle (Wde, Wde, Narrow) exsts f gushng probablty s n the range of 1.0 p The pure strategy does not exst f p 0.751; but, the mxed equlbrum strategy profle s (Wde, Don t drll, Wde). The pure equlbrum strategy profle (Narrow, Narrow, Narrow) exsts f 0.75 p If 0.49 p , the pure equlbrum strategy profle s (Don t drll, Wde, Wde). If p , the pure equlbrum strategy profle s (Don t drll, Narrow, Don t drll). If p 0.230, the pure strategy does not exst; but, the mxed strategy s (Don t drll, Narrow, Narrow). If p 0.001, the strategy profle for pure equlbrum of the game s (Don t drll, Don t drll, Narrow). If gushng probablty p = 0.00, the equlbrum strategy profle s (Don t drll, Don t drll, Don t drll). To sum up, Table 7 lsts all the stuatons. Table 7: The Gushng Probablty Range and Equlbrum Strategy Profle wth Type Gushng probablty Strategy profle (TAI,, HIB) Equlbrum profle 1.0 p (Wde, Wde, Narrow) Pure p (Wde, Don t drll, Wde) Mxed 0.75 p 0.50 (Narrow, Narrow, Narrow) Pure 0.49 p (Don t drll, Wde, Wde) Pure p (Don t drll, Narrow, Don t drll) Pure

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