Brain Drain, Brain Gain and Brain Return: possible ...

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using Nash's games theory it is possible to understand how to apply policies related ... Key Words: Brain Drain, Brain Gain, Brain Return, Migration flows, Skilled.
Brain Drain, Brain Gain and Brain Return: possible levels of equilibrium with the help of LISE or LSV Working paper Liminta Luca Giovangiuseppe1 (University Carlo Cattaneo - LIUC) March 2018 Abstract My theorethical paper aims to introduce a new possible level of equilibrium. With the help of my simple indicators LISE or LSV (Liminta Serati and Venegoni) that measure level of individual well-being linked to each country and it were perfectly comparable, I' m able to reach a new equilibrium or better a Nash's equilibrium instead a Pareto's optimal equilibrium which is impossibile to achieve with an empirical based model. With the help of my work it is more understandable which are the determinants of Brain Drain's equilibrium. If model's hypotheses involve a limited number of brains, then it is possible to approach Pareto's equilibrium but if the number of brains increases it falls to Nash's equilibrium. I have opened new avenues for research because, understanding that: the real equilibrium level is suboptimal and there is possible to reach a Pareto's optimal equilibrium only using a model with many restrictions. Policy makers who do not consider Nash's equilibrium as the only achievable can do a lot of damages trying to reach the optimum. With this study I 'm able to calibrate policies on a target equilibrium which would be really achievable.

1

Addresses: Liminta Luca Giovangiuseppe: Department of Economics, University Carlo Cattaneo LIUC, C.so Matteotti, 22, 21053, Castellanza (VA), Italy. email: [email protected].

using Nash's games theory it is possible to understand how to apply policies related to Brain Drain, Brain Gain and Brain Return. Key Words: Brain Drain, Brain Gain, Brain Return, Migration flows, Skilled migration, Skilled Workers, Games Theory, John Nash, Equilibrium. JEL Codes: F22, J61, O15.

1 Introduction

The goal of my theoretical paper is to introduce a new theory based on dominant strategy using some one-shot non-cooperative games to find some logical results, with the help of my well-being indicators. Using my indicator (LISE) (Liminta and Serati, 2016), I'm able to demonstrate that the presence of a dominant strategy brings the players to a sub-optimal equilibrium's level. With his Prisoner's Dilemma, Tucker attracted wide attention (A. W. Tucker, 1950) and his work helped me find a real achievable equilibrium level. My paper explains how brain drain results inevitable and why policy-makers must expect to achieve a suboptimal equilibrium's level. The first assumption of my work is that the game is played only one time, and this results to be a realistic hypothesis because migration is not an everyday choice. Moreover, my hypothesis that game is played once is near to reality because it is clear that: the approach used by people when they decide to migrate can't be similar to the decision to change a dress. The second hypothesis foresees imperfect information, as in Prisoner's Dilemma, the two players (countries) can't communicate. Country A cannot communicate with country B and vice versa. The third hypothesis is complete information on prize,

both players are offered the same deal, and they are completely aware that to the other player has been offered the same prize. Simultaneous game is my fourth assumption; without the possibility to communicate, players will make their decision at the same time. It is possible, solving the game, eliminating all dominated strategies to find a dominant strategy. 2 The solution of prisoner's dilemma model (A. W. Tucker, 1950) will bring to a Nash's equilibrium. 2 Literature review

Literature starts from the milestone of Tucker that introduced the Prisoner's Dilemma (A. W. Tucker, 1950). Following literature implemented that model with some possible solutions but, I assume that, considering a long period decisional structure (a country can take the decision, to implement or not its well-being, only one time every ten years to evaluate its effects), it is not possible to use a tit-for-tat strategy to solve the game, as done by Rapoport (Rapoport and Chammah, 1965). The game's type chosen is played only one time without the possibility, for 2

To design a prisoner's dilemma game we must order payoff: A > B > C > D. (For example A = 9, B = 8, C = 2, D = 0). We can simply observe that A > B and C > D. This is the payoff matrix: prisoner one lie

prisoner two

confess

lie B;B

A;D

D;A

C;C

confess

B;B is Pareto's optimal equilibrium but dominant strategy implies that the game will reach C;C that is a Nash's equilibrium.

competitors, to implement punishments. Previous research, also, underlined the importance to find a dominant strategy to solve one-shot games. I must clarify that, in my cases, it is not possible to consider a finite game played different times as written by Samuelson (Samuelson, 1987) because decision to implement LISE or LSV can be taken only one time. Previous contributions considered the possibility that length of the game significantly increases cooperation rates (Normann et al., 2005) and that we can find a low level of cooperation in games with a finite time horizon (Normann and Wallace, 2012), then, I consider, in my contribution, the game as played once: players have no possibility to cooperate. Another important assumption, to find a dominant strategy, able to solve the game, is that all players are rational, and they will choose their best solution. My last hypothesis is close to reality because policymakers, who decide for countries, must be rational; indeed, they must make a decision, to obtain the best prize for their country. In my paper, I have used previous findings to reach an important goal: dominant strategy brings to a Nash's equilibrium that will be distant to Pareto's optimal equilibrium (Nash, 1951). Nash was inspired by the model of Von Neumann and Morgenstern that proposes different solutions to the classical problem of bilateral monopoly (Neumann and Morgenstern, 1945). Previous authors have introduced the concept of "reputation" in finitely repeated game (Kreps and Wilson, 1982) and the problem that players don't play single period dominant strategy trying to achieve cooperation but, it is important to explain that, in my cases, decisions to migrate are not afflicted by reputation because too much time elapses between the decision to migrate and the next one. My work draws inspiration, even, by "The Tragedy of the Commons". In his work, Hardin, developing his idea from a W. F. Lloyd's tale (Lloyd, 1833), and with that, he discusses the possibility that the pursuit of

individual well-being can cause a social problem (Hardin, 2009). Snidal underscored the necessity to implement theory, with some auxiliary assumptions, to overcome its general structure. This author explains that a case study can provide a test of the explanatory power reached by the model because the theoretical constructs, related to the Prisoner's Dilemma, are flexible and they can be adapted to different types of problems (Snidal, 1985). Past literature, with the contribution of Docquier, was able to demonstrate, empirically, that poverty and lack of economic growth compel migrants to leave their countries. Obviously, my previous work introduced a simple, versatile and comparable indicator LISE that is able to simplify the way to reach a Nash's equilibrium whether it is brain drain or brain gain (Liminta and Serati, 2016). It is possible to use my simple indicator, inside a one-shot non-cooperative game, as a proxy of well being, to find a possible equilibrium's level.  

3 Nash’s games theory applied with the help of LISE or LSV

3.1 LISE and LSV cases (some important premises)

I have chosen to use a competitive game based on prisoner's dilemma model because, with the help of the explicative power of this model, it is simple to reach interesting logical results. Moreover, I can use my indicator LISE (Liminta and Serati, 2016) as a meter of well-being. My choice, of using a non- cooperative oneshot game, is made considering the long time horizon related to these kinds of migration's decisions. Nash based his work on the demonstration that

maximization of individual gain drives the system to a sub-optimal equilibrium point. This way of reasoning, merged to the prisoner's dilemma game, is perfectly helpful to explain migration processes. Including the possibility, for a country, to implement or not our indicator, it is possible to solve some cases with some simple theoretical solutions. It is important to underline that: LISE can be compared between countries simply as LISE A > LISE B or LISE A < LISE B. We accept as hypothesis, as demonstrated in my previous contribution (Liminta and Serati, 2016), that: at high level of LISE we can register a high number of brains. We must consider the possibility to implement LSV or not, as in LISE cases, and results reached are the same (high level of LSV means the presence of a high number of firms).   4 Using dominant strategy's theory with the help of LISE or LSV to prevent damages caused by brain drain

Two important hypotheses common to all cases are: 1. Countries make rational decisions as in prisoner's dilemma (A. W. Tucker, 1950). 2. Improving LISE also improves the attractiveness of a country, as demonstrated in my previous work (Liminta and Serati, 2016). I can show three cases using the theory of dominant strategy in some competitive games:  

Case 1 (LISE)

Assumptions are the same of prisoner's dilemma game's model (A. W. Tucker, 1950). I assume, as three more hypotheses: 1. The presence of only two countries 2. The attendance of only two brains 3. Inside my model I suppose the presence of one brain for each country and I consider a very small cost to improve LISE (Liminta and Serati, 2016) This is the pay off matrix (Table 1):   Country B does not improve LISE

improves LISE

does not improve LISE Country A improves LISE

1 brain, 1 brain

0, 2 brains

2 brains, 0

1 brain, 1 brain

Table 1

Probably brain drain doesn't occur or if it happens will be compensated by gain. No one country has the possibility to win the game. If the number of brains and the number of countries are limited, Nash's equilibrium can be very close to Pareto's optimal equilibrium (as in Table 1).  

Case 2 (LISE)

Assuming that principal hypotheses are the same of prisoner's dilemma (A. W. Tucker, 1950), I have introduced four new assumptions: 1. The presence of only two countries and n brains inside country A and m brains inside country B 2. The number of brains inside country A (𝑛) is different than the number of brains inside country B (𝑚) è n ≠ m 3. 𝑛   >  𝑚 4. I assume that is possible, for each country, to reach the same level of LISE and each country supports a very small cost to improve LISE (Liminta and Serati, 2016). These are results achieved (Table 2): Country B does not improve LISE

Country A

does not improve LISE

𝑛 brains, 𝑚 brains

improves LISE

𝑛 +  𝑚 brains, 0

improves LISE

0, 𝑛 +  𝑚 brains !  !  ! !

brains,

!  !  ! !

brains

Table 2   There will be only a winner: country B. On the contrary, the country with the highest number of brains will lose a part of its brains. Dominant strategy (  𝑛 + 𝑚   > 𝑛 >  

!!! !

country A will lose

  >  𝑚)   solves the game with a Nash's equilibrium and !  !  ! !

 brains. Country B, surely, will win the game, if it

participates, and country A, certainly, will lose the game. The country with a greater number of brains is obviously involved in the game but it can't win the match, A may only reduce damages. If we modify our initial hypothesis, considering a great difference in the number of brains, country a can lose a great part of its brains and, country B can greatly increase its endowment of brains (considering the possibility to reach the same level of LISE for each country).   Case 3 (LISE)

In this case, I consider a one-shot competitive game in which I suppose: 1. All hypotheses of prisoner's dilemma model (A. W. Tucker, 1950) 2. The presence of n countries and N3brains 3. I introduce a short case in which I consider two countries: inside country A a !

number of brains (h) higher (than the average ! ) and inside country B a lower !

number of brains (l) (than the average   ! ). 4. My last assumption is that: the other n countries improve LISE (but both countries, A and B, don't know this information), moreover, I consider a very small cost to improve LISE (Liminta and Serati, 2016) and the possibility, for every country, to reach the same level of LISE

3

The total number of brains is a strong limitation of my model because, especially in the United States, we have an elevate number of brains and, we can' t demonstrate that this model work, even if LISE is elevated and the number of brains is very high. I can solve this problem introducing a relative number or a percentage in my model and it will solve all problems. With the help of LISE's comparability, probably, these types of problems will not appear.

I can show two payoff matrixes, the first containing real payoff and the second one holding disclosed payoff (Table 3 and table 4):

Real payoff Country B does not improve LISE

Country A

does not improve LISE improves LISE

improves LISE

!  

 ℎ  brains, 𝑙 brains ! 5 !  !!

 brains, 0

0, !  !! 4    brains ! !

!

brains, ! brains

Table 3 (Disclosed payoff) Country B does not improve LISE

Country A

does not improve LISE

 ℎ  brains, 𝑙 brains

improves LISE

 (𝑁 brains)7, 0

Table 4 4

Real payoff Real payoff 6 Known payoff 7 Known payoff 5

improves LISE

0, (N brains)6 ! !

!

brains, ! brains

 

Country B must take a part in the game following the dominant strategy to get its disclosed gain (N) but the real prize reachable is:

!  !  !

!

+ !  !!   =   !  !  !

!

!  !  !  !  ! !  !  !

  =

!

 !  !  !  that, as supposed, is unknown and solving the game B gains !   trying to reach !

(N). Country B doesn't know that the real profit is !  !  ! but B will choose to improve its LISE even if this information is disclosed: B can only improve the total !

number of its brains because 𝑁   >  ℎ > !!!   > !

𝑙 <   !  𝑎𝑛𝑑

! !

! !

> 𝑙   and, as hypothesized

!

<   !!!. Country A must choose to improve its LISE too (Country A

doesn't know that there isn't the possibility to reach N) because there is the !

presence of a dominant strategy (𝑁   >  ℎ > !!!   >

! !

> 𝑙) that would bring it to !  !  !

!

lose a part of its brains. The real possible gain of country A is:   !  !!   + !  !  !   =  

!  !  !  !  ! !  !  !

!

  =   !  !  !, but it can't know this information. There is a dominant strategy !

(𝑁   >  ℎ > !!!   >

! !

> 𝑙)  that drives, clearly, both players to improve LISE and is

not possible for Country A to choose a different solution (even if the maximum !

prize was known, and !  !  ! < h, there no possibility for country A trying to reach h, without losing a great number of its brains). The possibility to country A to have a !

riservate information for example that the price is !  !  ! < h, can only damage A, because if A chooses "does not improve LISE", it will risk to lose all its brains in an open economy.

    Case 4 (LSV)

As in LISE cases my hypotheses are the same of the prisoner's dilemma (A. W. Tucker, 1950) with three more suppositions: 1. The attendance of only two countries 2. The presence of only two firms 3. I count one brain for each country and I have considered a very small cost to improve LSV   These are results achieved (Table 5):     Country B does not improve LSV

improves LSV

does not improve LSV Country A

improves LSV

1 firm, 1 firm

0, 2 firms

2 firms, 0

1 firm, 1 firm

Table 5

If the number of firms and the number of countries are limited, the equilibrium reached can be very close to Pareto optimal's equilibrium (as in table 14 an as in LISE case 1).

  Case 5 (LSV)

Assuming as valid all hypotheses contained inside prisoner's dilemma model (A. W. Tucker, 1950), I have introduced four more assumptions: 1. The presence of n firms inside country A and n firms inside country B 2. The numbers of firms inside country A (𝑛) is different than the number of firms inside country B (𝑚) è n ≠ m. 3. 𝑛   >  𝑚 4. I consider a very small cost to improve LSV and the possibility, for both countries, to reach the same level of LSV

This is the payoff matrix (Table 6):

 

  Country B does not improve LSV

Country A

does not improve LSV

𝑛 firms, 𝑚 firms

improves LSV

𝑛 +  𝑚 firms, 0

improves LSV

0, 𝑛 +  𝑚 firms !  !  ! !

firms,

!  !  ! !

firms

Table 6   Even in this LSV case: country A that contains a higher number of firms lose the game, hence, A can only limit damages converging to average. Dominant strategy is improving LSV for each country (𝑛   +  𝑚   >  𝑛   >  

!  !  ! !

  >  𝑚). B will improve

its endowment of brains winning the game. This game reaches a Nash's equilibrium that it is sub-optimal.  

Case 6 (LSV)

In my last case, I consider a one-shot competitive game in which I suppose: 1. I assume all hypotheses related to prisoner's dilemma as valid (A. W. Tucker, 1950) 2. n countries and N firms

3. I introduce a short case in which I consider two countries: a country A with a higher number of firms (h) higher than the average and a country B with a lower number of firms (l) than the average. 4. The other n countries improve LSV (both countries, A and B, don't know this information) because I consider a very small cost to improve my indicator and I consider the possibility, for every country, to reach the same level of LSV  

I can show two-payoff matrixes, real and disclosed (Table 7 and Table 8) Real payoff Country B does not improve LISE

Country A

does not improve LISE improves LISE

 ℎ  firms, 𝑙 firms ! 9 !  !!

 firms, 0

Table 7

 

8 9

Real payoff Real payoff

improves LISE

!  

0, !  !! 8  firms ! !

!

firms, ! firms

(Disclosed payoff) Country B does not improve LISE

Country A

does not improve LISE

 ℎ  firms, 𝑙 firms

improves LISE

 (𝑁 firms)11, 0

improves LISE

0, (N firms)10 ! !

!

firms, ! firms

Table 8   !

Country B is dominated by strategy to improve LSV ((𝑁)   >  ℎ > !!!   >

! !

>

𝑙)  but the strategy of Country A, to maximize its gain, is to reach N too (because A has not a complete information about the strategy of the other countries as in "3 LISE case"), hence, I can conclude that, solving the game, with the help of the dominant strategy, countries with an higher (than the average) number of firms can only provide to limit damages improving their indexes and countries with a lower (than the average) number of firms can only pick up their gain improving LSV.  

 

 

   

10 11

Known payoff Known payoff

5 Conclusions

The assumptions of prisoner's dilemma model are particularly adaptable to brain drain or firm drain because the choice to migrate can occur only one time in four or five years, for people, or in minimum ten years, for firms. Choosing a one-shot game appears as a correct choice, especially if we consider a vast time horizon, which occur between decisions. The important goal of my work is to demonstrate, with the help of prisoner's dilemma theory, that countries with a stronger endowment of brains can always lose a part of its brains, and countries with a less number of brains are incentivized to improve their attractiveness, for the reason that, they can only improve their number of brains. For these reasons, countries with a higher number of brains should find an agreement with countries that show a lower endowment of brains but dominant strategy leads countries (with a lower endowment of brains or firms) to improve LISE or to improve LSV to attract to itself more brains or more firms. My work will have a great relevance for policymakers because they can make macroscopic mistakes, without a proper interpretation of the phenomena related to firms or high-skilled migration flows. The dominant strategy is fixed, by the presence of outsider countries that have the necessity to implement their attractiveness, to take their gain, and by the possibility for countries, with a larger endowment of brains, to increment their indicator to improve their gain. The best performers can only minimize damages to themselves. Obviously, my findings have two strong limits: one limit is represented by the possible strong difference between the index of country A and the indicator of country B. If there is a strong difference, increasing LISE or LSV will be not

sufficient to attract migration's flows, especially in the short period. The second limit is explained by globalization: migrants become citizens of the world and the game could be repeated more frequently, hence, we could implement, in the future, a model based on bargaining problems. These conclusions will help policy-makers to prevent strong damages to their countries or to implement new policies to attract high-skilled migration flows. Implementing a complete policy is the only possible way to prevent a lot of strong economic damages. Losing high-skilled persons is, as verified by previous literature, a big deal for each country.

References

A. W. Tucker, 1950. A two person dilemma. Stanf. Univ., Mimeo. Hardin, G., 2009. The Tragedy of the Commons. J. Nat. Resour. Policy Res. 1, 243–253. https://doi.org/10.1080/19390450903037302 Kreps, D.M., Wilson, R., 1982. Reputation and imperfect information. J. Econ. Theory 27, 253–279. https://doi.org/10.1016/0022-0531(82)90030-8 Liminta, L.G., Serati, M., 2016. Brain Drain, Brain Gain and Brain Return by Luca Giovangiuseppe Liminta - Research Project on ResearchGate [WWW Document]. ResearchGate. URL https://www.researchgate.net/project/Brain-Drain-Brain-Gainand-Brain-Return (accessed 5.23.17). Lloyd, W.F., 1833. The Notion of Value. McMaster University Archive for the History of Economic Thought. Nash, J., 1951. Non-cooperative games. Ann. Math. 286–295. Neumann, J. von, Morgenstern, O., 1945. Theory of Games and Economic Behavior. Princeton University Press. Normann, H.-T., Wallace, B., 2012. The impact of the termination rule on cooperation in a prisoner’s dilemma experiment. Int. J. Game Theory 41, 707–718. https://doi.org/10.1007/s00182-012-0341-y Normann, H., Wallace, B., C, J.C., 2005. The Impact of the Termination Rule on Cooperation in a Prisoner’s Dilemma Experiment. Rapoport, A., Chammah, A.M., 1965. Prisoner’s Dilemma: A Study in Conflict and Cooperation. University of Michigan Press. Samuelson, L., 1987. A note on uncertainty and cooperation in a finitely repeated prisoner’s dilemma. Int. J. Game Theory 16, 187–195. https://doi.org/10.1007/BF01756290 Snidal, D., 1985. The Game Theory of International Politics. World Polit. 38, 25–57. https://doi.org/10.2307/2010350