B Zimmermann

J J Chanaron

L Klieb

A Benchmark for Managerial Effectiveness

Keywords:

This study presents a tool to gauge managerial effectiveness in the form of a questionnaire that is easy to administer and score. The instrument covers eight distinct areas of organizational climate and culture of management inside a company or department. Benchmark scores were determined by administering sample-surveys to a wide cross-section of individuals from numerous firms in Southeast Louisiana, USA. Scores remained relatively constant over a seven-year timeframe. Techniques for using the benchmark by practitioners are discussed.

Introduction and overview

1.1. Overview An important part of the task of managers is to influence and steer the climate and culture within their organization so that the goals of the organization are reached as efficiently and effectively as possible. It would seem natural that managers would always recognize their employees' feelings and thoughts. However, counter-intuitively, many managers only have a very faint idea how the culture and climate of their organization is perceived by their employees.

Different organizations have different cultures, and differences in cultures do not necessarily dictate success or failure in reaching some goals of an organization, like profitability or expansion. Nevertheless, a negative climate can also sometimes have disastrous consequences for reaching the goals of the organization. It can decrease productivity, increase absences and sick leaves, lead to high turnover and churning, and have many other negative influences. Managers are seriously lacking if they do not take their employees' personal needs into account [START_REF] Cangemi | North American employee attitudes in the 1900's: changing attitudes changing times[END_REF]Bailyn, Fletcher, & Kolb,1997).

Perceptions of culture and climate within an organization can change quickly. Change is inevitable. Managing change can be studied and analyzed from many different viewpoints [START_REF] Schein | Organizational learning: what is new?[END_REF][START_REF] Jacobs | Real time strategic change[END_REF]. The results of the present study provide managers with a new tool, the Management Perception Survey, to compress greatly the time and effort required to gauge adequately and accurately the climate and culture of their organization and the possible need for change.

The Management Perception Survey (MPS) is designed to paint a picture or mosaic of the state of the organization's culture and climate as it is perceived by workers and management. It provides a snapshot at a given point-in-time. It can serve as tool for management to increase its effectiveness in influencing climate and culture. Therefore, the results of the survey function also as an indirect measure of management's performance of its on-the-job responsibilities for matching the needs of its employees and of the company.

The MPS questionnaire consists of a beginning section with instructions and questions devoted to the demographics of the respondents. This is followed by eight individual construct sections, referred to herein as Survey/Constructs, and titled Survey One through Survey Eight. Each of these eight Survey/Constructs or sections contains twenty statements about: Morale and Motivation; Training and Evaluation; Human Relations; Communication; Resources; Problem Solving Strategies; Managerial Stress and Equality and Diversity. These constructs are directly associated with the following five functions of management that have to do with interactions with employees: Planning, Organizing, Directing, Staffing, and Co-ordination [START_REF] Gulick | Papers on the Science of Administration[END_REF]. The other two of the seven "PODSCORB" principles, Reporting and Budgeting, are not addressed in this study, as they do not pertain to employee interaction.

Each Survey/Construct contains 20 questions. Ten questions draw from policy and procedure, and ten questions to the actions and interactions of the employees at all levels.

The questions are formulated as a grading exercise. The respondents are asked to grade their organization, management, and coworkers on a scale from 0 to 10 on an aspect of how the organization functions. In order to arrive at a benchmark number, the scores of each of the eight surveys are totaled and divided by 2. This gives an outcome between 0 and 100 for each construct. An overall grade is obtained by averaging the results of the eight surveys. The full questionnaire can be requested from the first author by email.

Besides introducing the surveys, this study provides results from samplings carried out in Southeast Louisiana in 1995/1997 and in 2002. The results of these samplings indicate the validity and reliability of the instrument. The study also provides benchmark numbers that practitioners can use to compare their own results with a cross-section of industries in this area.

This study reinforces the theory that technologies and goals may change from industry to industry, yet people will tend to react to positive or negative management behavior in a similar fashion regardless of the industry [START_REF] Maslow | Motivation and Personality[END_REF][START_REF] Maslow | Toward a psychology of being[END_REF][START_REF] Maslow | The far reaches of human nature[END_REF]Herzberg, 1972[START_REF] Herzberg | One more time: how do you motivate employees?[END_REF]McGregor, 1972, Zimmermann, 1998 p. 205).

There are several ways to use the benchmark. Managers can administer the surveys in their organization, or outside consultants can use it to determine quickly the state-of-affairs within an organization. The numerical results of the eight Survey/Constructs give a quick impression of how the organization fares in any particular area. Low results of a particular grouping of questions point to a problem in that area. This can then be analyzed in detail to gain a better understanding of what the issues are.

Research process and design

In 1989, the initial questionnaire was created. It was based on pragmatic personal management experience, combined with a study of the academic literature in the leadership and management areas including such topics as human needs, morale, motivation, leadership, supervision, training, planning, and communication. The resulting MPS provided a quick and effective way for learning how the employees perceived certain aspects of their management's effectiveness.

This research project has been continued since 1989, with only minor improvements and enhancements to the sample-survey instrument itself. Originally, it was only envisioned as a snapshot of one particular organization at one particular time. However, many of the individuals who have taken part in this study have asked the following two questions:

• How does the management effectiveness score for my company compare to the scores of other companies?

• •
Why is my company's overall grade better or worse than the score other companies get from their employees?

• These two important questions, the first one asking how do I compare, and the second asking what the comparison means, prompted research to see if a relevant general benchmark, with stable scores over time and over various industries, could be derived by sampling and statistically testing a sufficient number of employees. The answer turned out to be affirmative. The uniqueness of this study lies in the effort to gather data from many industries located over time within a welldefined geo-social area.

Discussion

This study is not intended to explain why employees feel the way they do about their organization's managerial effectiveness, climate and culture. Quoting [START_REF] Sabourin | A matter of perception[END_REF], "A perception survey measures how the people perceive the situation, it records what they believe to be true, and it gauges opinion, outlook and attitude. In other words, it measures the 'culture' or the operative reality in the workplace."

Employees' perceptions of the strengths and weaknesses of an organization are not necessarily an indicator of the level of performance of the organization. In addition, depending on the circumstances, perceptions can change rapidly. However, the general agreement among most management practitioners is that employee perception of what the reality is in the environment of that workplace influences their behavior [START_REF] Kienzle | The relationship between organizational climate and employee perceptions of involvement[END_REF]. Blacker (1984) wrote that regardless of the actual facts surrounding any given situation, perception is reality to the perceiver.

This study addresses the reaction of employees to various approaches to the management of technology involving both people and operations. Technology Management and the Management of Technology are integral functions of business today [START_REF] Chanaron | Technology management: a tentative research agenda[END_REF]. The information gathered by the MPS provides internal managers as well as outside consultants with a valid roadmap for identifying both positive and negative issues at the time the survey is conducted in an organization.

In the end, however, the real value in finding out how employees feel and think is realized only if something is done about it. If management is not really interested in finding out how their employees view their work environment, then it is a waste of time to go to the trouble of conducting this type of survey (Altmann, 2000).

Although there are common elements in many perception surveys, the MPS is different from other perception surveys in the literature. It is not intended as a tool for an overall productivity or outcome performance evaluation (Green & Henderson, 2000), nor as a definitive gauge of employee job satisfaction [START_REF] Janssen | Fairness perceptions as a moderator in the curvilinear relationship between job demands, and job performance and job satisfaction[END_REF]. The purpose of this study is to show that its use as a diagnostic tool is practical and valid. It provides timely data that is useful for management.

Construction of the Questionnaire

The definition of Benchmark used for this study is: "A standardized problem or test that serves as a basis for evaluation or comparison" (Merriam-Webster, 2004).

In this study, the following five functions of the PODSCORB management tasks are assessed: Planning, Organizing, Directing, Staffing, and Coordination. The MPS seeks data concerning all three levels of internal stakeholders within an organization: executive policy administrators, mid-level to senior management personnel, and the operational workers. A basic tenet of this study proposes that employees react to management in a similar fashion in all types of businesses. The basis for the contents and format of the data gathering sample-survey questionnaire are stated in the two hypotheses presented below: Hypothesis #1) Any company's overall organizational management effectiveness can be broken down into a limited number of areas of study, which encompass the major aspects of organizational behavior, culture, and managerial effectiveness We will use eight different areas here and exclude non-humanistic aspects like finances and capital resources.

Hypothesis #2) Any company's perceived overall people management skill level and organizational effectiveness is functionally independent from that company's actual physical and financial resources.

In other words, the study assumes that the basic functions and skills required for effective human resource and/or organizational management are fundamentally similar for all organizations, regardless of the number of employees; the size of a company; the financial strength or profitability of the organization; or to the type of industry or profession.

Although theoreticians have expanded on and offered variations explaining employee behavior and motivation, the theories proposed by Maslow and others still provide a sound framework [START_REF] Maslow | Motivation and Personality[END_REF][START_REF] Maslow | Toward a psychology of being[END_REF][START_REF] Maslow | The far reaches of human nature[END_REF]McGregor, 1972;[START_REF] Herzberg | One more time: how do you motivate employees?[END_REF]. In their vision, managers and business leaders are responsible for creating a working atmosphere that meet most of those basic human needs in a manner acceptable to the employees. The concept of meeting the needs of employees forms a solid foundation for each statement contained in the MPS.

The following is a description of how the Benchmark was constructed. The following eight areas, considered closely tied to Planning, Organizing, Directing, Staffing, and Coordinating, were identified as factors that influence management's effectiveness in influencing the culture and climate of an organization: In each area above, ten questions were constructed that measure the influence of policies and procedures within the company on culture and climate in that general area of concern, and ten questions gauging the actions and interactions of the employees at the worker and manager level. The latter were sub-divided into five statements concerning the line-workers and five statements involving the working managers.

Questions were formulated in the form of a positive statement pertaining to the subject matter of the individual construct. In that way, the formulation of the question did not suggest any negative feeling or connotation to the respondents. As a sampling control measure, each of the eight Survey/Construct sections or scales contain statements which are similar in context to statements on one or more of the other Survey/Construct sections. The Zero (0) to Ten (10) scoring scale was used to mimic the normal range of grading familiar to most employees who have been educated in elementary, secondary, and higher education institutions in the United States. [START_REF] Keller | The Brand Report Card[END_REF], in his Harvard Business Review article discusses benchmarking brands, and uses a survey constructed in a similar way to the MPS, with a scale from One to Ten and all statements worded positively.

The results of the MPS create a clear picture of how the employees perceive the effect of their management and fellow workers on the culture and climate of the organization at a given point in time. It is analogous to the Financial Balance Sheet, as opposed to an Operating Statement. The MPS equates to the Balance Sheet, while causal studies correspond to the Operating Statements.

This study does not propose a specific causal relation with an outcome. It is different from other studies which seek to establish a link between specific behavior and specific outcomes. This is not a limitation but a strength: independent of the causes, we show that employees judge management on the same things.

By using a quantitative sampling method instead of an interview-based study, the timeframe for evaluating the state-of-affairs of a firm is reduced from weeks or months to at most a few days, and the results provide more detailed and standardized information than would be gathered from personally interviewing and observing.

According to a study of 523 abstracts by [START_REF] Nosek | E-research: ethics, security, design, and control in psychological research on the internet[END_REF], the most prevalent research themes in leading management journals are: Leadership, Strategic Management, Customers, Information Analysis, Human Resources, Process Management, Business Results, Implementation, and Evaluation and Assessment. The MPS contains statements that directly or indirectly relate to all of these content areas.

Related Literature

There is scant literature involving studies as wide ranging in content or which has been undertaken across as many types of industries as this study. Examples of questions in the eight areas can be found in the Appendix. The statements in the MPS concerning managerial effectiveness, leadership, communication, organizational climate and culture conform to generally accepted theories in the scientific literature. They reflect conditions that much of the current management authors consider essential for good management in the eight constructs listed. We refer to [START_REF] Bain | Are you underestimating your team members?[END_REF][START_REF] Denning | Find ways to motivate your staff and give morale a boost[END_REF][START_REF] Baldwin | Is anybody listening? Performance evaluation feedback in the U.S. air force[END_REF][START_REF] Bettenhausen | Peer reviews: employees' dual roles as rater and recipients[END_REF]Altmann, R., 2000;[START_REF] Ben-Ari | Walking the tightrope between work and family[END_REF][START_REF] Clement | M&A's: preventing culture clash[END_REF] Especially within the United States, organizational attitude surveys have become very popular within the corporate culture. The 1999, the Keinzle, Rodwell, and Shadur study identifies three essential variables that are most significant in studying the perception of employee involvement: 1) Input In Decision Making, 2) Teamwork, And 3) Communication. The Keinzle et al. study also looked at employee perception of job satisfaction, commitment, stress, bureaucracy, innovation, and support. The results show that supportive environments and commitment were predictors of each of the employee involvement variables (Keinzle, et al., 1999[START_REF] Pekala | Employee reviews -perform at your own risks[END_REF] reports on an interview discussing the use of performance evaluations with Diane Kristen, Esq., principal of Kristen Coburn Chartered. Kristen points out that the main problem with individual performance evaluations is that they are totally dependent on preconceived notions and prejudices of the evaluators. Likewise, with this study, the opinion of one individual regarding the managerial effectiveness of a company can be influenced by the biases and disposition of that employee. However, the perceptions of a sufficiently large group of employees indicate a general feeling towards the organization including any group dispositions. In order to get significant results, we suggest that the questionnaire is administered to each member of a department (inside a large company) or the whole company if sufficiently small.

Benchmarking is generally accomplished by making comparisons of performance of organizations within a given industry. The MPS does not look directly at performance numbers.

Instead, it looks at the scores to questions on the human support side of five functions of management. Regardless of whether performance numbers are good or bad, by carefully discerning the meaning of the data supplied by the MPS, management has a better chance of understanding the reasons for performance output numbers.

Summarizing the literature, there is ample evidence that using employee perceptions for evaluating the management effectiveness and organizational climate and culture is widely accepted within the academic and business arenas. None of the previous studies reviewed for this study attempted to establish a cross-industry benchmark to establish a scale for management effectiveness, climate and culture. Knowledgeable managers can use the results of the MPS to help determine the strengths and weaknesses of their organizations.

The research instrument takes the place of the interviewer, and sets a scale of numeric parameters used to answer the questions. It could be described as a paint-by-the-numbers picture of the organization, with the respondents as the artist. However, just as the qualitative researcher must interpret what interviewees say, so, too, must the observer interpret the story told by the scores of the various statements in the MPS.

Setting the benchmark: Sampling Methodology

The primary quantitative research to establish a benchmark involved administering the MPS to a cross-sectional, cross-industry sampling of employees.

The primary research sampling in 2002 was accomplished using the Internet to distribute and collect Excel based electronic questionnaires via E-mail. The sample-surveys were e-mailed to specific pre-identified respondents who had expressed an interest in participating. These respondents were both undergraduate and graduate students enrolled at night in a regionally accredited university in the Southeast of the USA. They were all at least twenty-three (23) years-of-age and worked full-time at various organizational levels within their companies, including: lineworkers, low to mid-level managers and senior executives. The average age of the students was thirty-four (34) years of age. Students represented a range of organizations of various sizes in multiple industries located in Southeast Louisiana. They were advised that the data would be used in a research project for a doctoral dissertation. Completion of the survey was voluntary, not a requirement for course grades, and students were not paid for their participation. The respondents were advised that the research did not seek their views of the University as students.

Most respondents reported that it took them approximately 20 to 25 minutes to complete the sample-survey. The problems of representation and generalization do not affect this study since the use of the Internet was restricted to specific pre-identified participants.

The questionnaire used in the sampling portion of the data gathering process was designed to gather information in two general areas:

1. Demographic information about each individual respondent who completed the samplesurvey. 2. The respondent's personal perception of the organization's management and leadership effectiveness.

We analyzed then our data and found that scores were generally independent of aspects of demographics like firm size, industry, and position inside the company (with some exceptions to be discussed below) and stable over time. This establishes our claim that we have established a tool that can function as a benchmark.

The current electronic version, titled Perception Survey in Excel, contains 160 statements, divided into 8 groupings or constructs of 20 statements each, referred to herein as the Survey/Constructs. The respondent simply checks-off the blank space for the demographic option that fits his or her situation, and enters a number from 0 to 10 for each of the 160 statements.

Results of the Statistical Testing for the 2002 Survey

The computer program SPSS was used to analyze the data. The tests considered the most significant for establishing validity are presented in this article, namely: General Descriptives, Internal Reliability (Cronbach's Alpha), and the One-Sample and Two-Sample Kolmogorov-Smirnov tests.

For our sample of 117 sample-surveys, we found a high degree of Internal Reliability in all of the eight survey/constructs according to a measurement of Cronbach's alpha. Taking into account the number of questions, these results indicate that the average correlation coefficient between each of the 20 questions of the Survey/Constructs is high. Cronbach's alpha for the overall score as an average from the 8 Surveys/ Constructs is high as well. Numbers are presented in table I INSERT TABLE I ABOUT HERE Some demographic questions were asked for this sample-survey. 101 out of 117 respondents answered the question "How long have you been a people manager" . 10.9% indicated that they had been in management for less than (3) years; 32.7% answered between (3) and ( 10) years; and (56.5%) indicated they had been in management for over (10) years. Forty-two respondents stated they were not in management. The breakdown of the "How many people were managed" category was 36.5% managed less than (5) people, 25.2% chose between ( 5) and ( 10) people; and 38.3% claimed that they managed more than (10) people.

Forty-three and six-tenths percent (43.6%) of the respondents signaled that they were experienced in management when they were hired, while 56.4% said they were not experienced in management at the time of hire in their present job.

When asked if they were "promoted from the ranks' when first appointed to management, 50.5% responded YES and 49.5% said NO.

Only 32.4% responded YES to the question regarding the formal training prior to being appointed as a manager, while 67.6% answered NO to that question.

The percentage of respondents answering YES to the question asking if they had received in-house or on-the-job training was 53.6%, with 46.4% checking NO.

When asked their perception of profitability or efficiency of the organization, 3.4% thought their organization was Highly Profitable or Efficient, 5.2 % judged it Moderately Profitable or Efficient; 15.5% considered the organization to be Marginally Profitable or Efficient; 38.8% offered Barely Profitable or Efficient; and 37% thought the organization was Unprofitable and/or Ineffective and Inefficient.

In its simplest terms, the Benchmark is expressed as the Mean and Standard Deviation for the eight survey/constructs and an overall average, like in Table II:

INSERT TABLE II ABOUT HERE

For each of the eight Surveys/Constructs, a score of around 70 is average, but the spread is fairly large among the various respondents. Therefore, the mean provides a very quick indication to the sample-survey-taker how the company is doing. A "deep" statistical analysis is unnecessary to administer or interpret this instrument. A simple calculation of the mean score from the twenty questions for each construct is sufficient. Overall scores per Survey/Constructs far below 70 are a sign of potential trouble. Scores substantially higher than 70, indicate that the perceptions of the members of the organization are exceptionally positive. Note that, on a scale of 0 to 10, 7 (a "C" in school terms) is considered, by most individuals, the "average" academic grade that is sufficient to pass without distinction and with a reasonable but unremarkable performance.

The fairly large number of twenty questions per construct leads to an approximately normal distribution for each of scores of the eight survey/constructs. This was checked non-parametrically with the One-Sample Kolmogorov-Smirnov test assuming a normal distribution. The results for the overall score and most of the eight constructs are normally distributed. If they are not completely normal, then they are very close to normal. Table III below gives the asymptotic sigma (2-tailed) value assuming a normal distribution for the non-parametric One-Sample Kolmogorov test to test for a normal distribution:

INSERT TABLE III ABOUT HERE

Only construct 5 (and 3 up to a degree) deviates slightly from a normal distribution, and even those have a Bell-curved shape. The individual question with the lowest overall score among all 117 respondents in this 2002 sample-survey is "The organization is fairly problem free", which is part of the Survey 6, Problem Solving. This question gets an average score of 5.2. It seems that rarely people consider their organization problem free, which is understandable considering the challenges in nearly every business. The highest average score for any question among the 117 respondents was 8.4, for "The organization believes in service", on Survey Three (Human Relations). This particular result may reflect the relative preponderance of service-oriented industries in Southeast Louisiana and supports the possibility that a benchmark like this one should be calibrated by comparison with industries in the same geographical area.

The median score is 6.80, out of the 160 questions, 80 got an average score of below 6.80 and 80 above. The answers to most individual questions are not normally distributed, and for the purpose of this study they don't need to be, and even shouldn't be. The questions have been deliberately and carefully constructed in such a way that they discriminate between different environments. Consequently, for a few questions, answers cluster around an average value and the answers follow a normal distribution. Some questions are polarizing, and the answers are multimodal: respondents give organizations either a low score or a high score, with few answers scoring in-between. Some other questions are generally given very high scores, but the respondents who give (very) low scores indicate that something is (seriously) remiss in this area in their organization. An example is the question "I feel that I am accepted and treated very well by my fellow workers", which is question 20 in the Managerial Stress survey (Survey Seven). Here 29.1 percent of the respondents gave this question a grade of 8, 29.1% of 9, and 16.2% of 10. So nearly 75% answer this question very high. An additional 11.1% give this an (average) score of 7. The 15% of the respondents that give a (very) low score clearly are expressing their dissatisfaction with an important part of their work environment. Statistically, the skew-ness of this question is -1.837 (standard error of 0.224).

The lack of a normal distribution for the individual variables was confirmed nonparametrically with the one-sample Kolmogorov-Smirnov distribution, assuming a normal distribution. For nearly all questions the 2-tailed asymptotic sigma distribution as zero or very close to zero, indicating a very bad fit to a normal distribution. However, the range in answers for all questions is quite high, by construction. All questions have a sufficient range in scores to make it worthwhile to include them in the sample-survey. Standard deviations (still useful as a measure of dispersion even for non-normal variables) ranged from 1.69 for the question "The overall job efficiency of the workers is excellent" (which clearly did not lead to large outliers and was not seen as very controversial) to the question that generates the largest range in reactions: "The managers never criticize other managers behind his or her back", with a standard deviation of 2.81. The latter question is part of Survey Eight, Equality and Diversity. Respondents give their organization a larger variation in grade in this area. Clearly, this is just an example of how large the differences in culture can be between organizations. This benchmark serves as a quick diagnostic tool to point out those differences in many relevant areas.

As also can be seen from Table 3 above, Survey Eight, Equality and Diversity, has in general the widest range in scores. This is interesting because this survey only asks very general questions, like "The organization treats all employees like people and not as objects", in order to gauge how the organization values diversity, differences of opinion, etc. This sample-survey does, deliberately, not ask any particular question about explicit racial preferences, about perceived discrimination, etc. Nevertheless, it has been our experience by collecting qualitative information that it gauges perceptions that are relevant in these areas accurately. This confirms again the central hypothesis of this research, that different organizations differ greatly in their organizational culture and in how far they satisfy the needs of the people that make up the organization.

It is important to realize that the lack of normal distribution does not reflect an error in scale construction. A normal distribution is to be expected when many independent smaller variables contribute to one dependent variable. In this study the questions have been formulated in such a way that they are as uni-variate as possible.

Because of the non-normalcy of the variables, we have not been able to do a post-hoc factor analysis to see if independent factors contribute to the eight postulated constructs among the 160 questions. Principal component analysis relies on factoring out first the factor with the largest variance. If the distributions of the variables are not normal, then the goal-seeking part of the factor analysis algorithm breaks down.

Obviously, a good instrument minimizes its dependency on demographics. A one-way ANOVA analysis did not find any statistically significant dependency of the results of the eight constructs on demographics (like stated above, those are normally distributed). However, it is possible that there are certain clusters of questions that depend stronger on demographics, but those have not been identified.

In summary, we have shown the following for the results of the 2002 survey:

• Internal Reliability of the Scales is excellent • There is a large spread in demographic information among respondents, and the benchmark does not depend on the demographics • The 8 scores for the survey/constructs display a normal distribution among respondents • Most individual questions do not have a normal distribution. All questions have a sufficiently wide range of answers among respondents • The benchmark is statistically valid 2.5. Results of the Statistical Testing for the 1995 to1997 sample-surveys. This sample-survey was carried out among 85 adult students in a classroom setting. The demographics of those students were mostly identical to the demographics of the students participating in the 2002 study. Since this was a post-hoc analysis of the data from the original documents filled out by hand, the respondents' demographics were of no interest and not statistically tested. The results of this sample-survey were remarkably similar to the 2002 surveys, supporting the premise that this benchmark is capturing essential information about the climate and organization of an organization. Survey Eight on Diversity was not in this survey but was introduced later. The results of the first seven Survey/Constructs can be compared directly, as the questions were the same. However, care should be taken when the overall score is compared, because in the 2002 survey this is an average of 8 Survey/Constructs and in 1995/1997 an average of only seven.

Like the 2002 results, Cronbach's alpha was high, indicating excellent internal reliability for the various scales. In summary:

• Internal reliability was excellent and comparable with the 2002 survey

• Results for the 1995-1997 survey were very similar to the 2002 survey

The results of the Cronbach's alpha and the Mean and Standard Deviation of the earlier surveys are shown in Tables IV andV below.

INSERT TABLE IV ABOUT HERE INSERT TABLE V ABOUT HERE

Comparison of 1995-1997 and 2002 surveys

The results for Means and Standard Deviations for the 1995/1997 and 2002 surveys are very close. For the original seven Survey/Constructs, they are generally all around a score of 70, and within a 2.5 range of each other, sometimes closer. We used two different, independent ways to confirm this result. When the distribution of variables to compare is normal, the t-test for two independent samples can be used (and the test is fairly robust with regards to this assumption). We have seen before (table 3) that the distributions of the different Survey/Constructs are close to normal. First Levene's test for equality of variances confirmed that there was no statistically significant differences in the variances shown in tables 2 and 5 above. Values of sigma for this test for the 7 Survey/Constructs were 0.708, 0.887, 0.838, 0.179, 0.707,0.327, and 0.701. Therefore we assumed that variances are equal. With this assumption, there is no statistically significant difference between the means of the seven Survey/Constructs. The t-test for equality of the Mean resulted in values of sigma (two-tailed) of 0.686, .660, .283 , .333, .961 , .563, and .269, confirming that there is no statistically significant difference between the results for the 1995/1997and 2002 surveys.

Alternatively, one can assume that is nothing is known about the distributions and apply the two-independent-sample Mann-Whitney test, a non-parametric test. The results for the overall score (for seven constructs) and for the separate seven Survey/Constructs all indicate that the test does not find any statistically significant differences between the 1995/1997and 2002 surveys.

The same test was applied to all 140 variables of the first seven Survey/Constructs. Remarkably enough, even at this granular level, most of the variables did not show a statistically significant difference between the 1995-1997 and 2002 surveys! Out of the 140 variables, only 33 had a two-tailed sigma for the Mann-Whitney test below 0.1 (indicating some level of disagreement), and only 18 out of the 33 had a two-tailed sigma for this test below 0.05 (indicating a strong difference).

Although this was not a true longitudinal study, the comparison between 1995/1997 and 2002 does have a quasi-longitudinal aspect. The similarity between the data collected in 1995/1997 and 2002 from different students provides a clear indication that the humanistic psychological and geo-social influences on the average perceptions of management effectiveness in Southeast Louisiana has not significantly changed over the years. In summary:

• Questions were answered in a similar way in the two surveys from 2002 and 1995-1997 • Benchmark results for the 7 out of 8 survey/constructs that could be compared were statistically not different 2.6. Other surveys From 1997 until 2000, 185 surveys were administered to a similar group of students. Results for average scores and standard deviations were not statistically different from the other studies. No further analysis was carried out.

The instrument: use and limitations

The use of the instrument

In 1990, the first author used the MPS to analyze the conditions within the management ranks of a major division of a billion dollar corporation located in Southeast Louisiana. This division had been losing millions of dollars every year for several years. The sample-surveys were administered during an all-day training seminar and the results discussed as the day progressed.

The scores, shown in

INSERT TABLE VI ABOUT HERE

The scores on Morale and Motivation and Human Relations showed that morale of the midlevel and senior managers was very low. They felt that they were being blamed for the losses of their division. All of the reasons for the low scores were discussed during the seminars.

The MPS was subsequently used in a similar fashion during a management-training seminar in a local steel fabrication company also located in Southeast Louisiana. In this case, as seen in Table VII, the resulting scores fell more closely in line with the Mean benchmark. In this second case, the MPS was also completed by the two top executives of the company. The Overall Mean results of their sample-surveys is also shown in Table VII below.

INSERT TABLE VII ABOUT HERE

The grade awarded by the President is substantially higher than the average grade given by the employees, and is an illustration of the concept of "Disconnect" that sometimes occurs between what management perceives and what the employees perceive.

During both of these seminars, low scores on statements such as Survey Three, Statement #12, "The workers trust the managers" were discussed in detail in order to find the root cause of the low grade. After both of these management-training seminars, a detailed written report of the findings and implications was submitted to top management.

The electronic version (Excel spreadsheet) lends itself to this same type of review and discussion, with the advantage being that the sampling can be done ahead of time, and quickly. This gives the manager, trainer, or consultant more time to analyze the results and prepare to discuss them.

Limitations and further research

The Mean and Standard Deviation benchmarks, along with the descriptive statistics derived from this study, provide strong evidence in support of the concept of this study. But this study is also limited in scope in some respects.

• Results have only been collected for organizations located in a limited geographical area (Southeast Louisiana, USA) • It does not address potential underlying political or psychological influences on the manner in which employees perceive management • It leaves room for studying both the employees' and the organization's financial condition and the financial effects on the scoring • It is more effective in smaller organizations • Use of the electronic version may be hindered if there is a lack of computers with some employees, and completing the survey by hand is more time consuming.

The results of this study prompt us to give suggestions for five main areas of further research involving the general concept of gauging employee perception of management's effectiveness. These include:

• Possible geo-social variations in the benchmarks • Causal relationships between the benchmarks and performance outcomes,

• Changes in the questionnaire design • Scoring • Drivers of employee expectations • Discussions of these four main areas of further research follow.

Geographical and Geo-social Expansion

One of the limitations in this study is the restriction of sample data gathering to organizations located within the geographic area of Southeast Louisiana. Completing the research in other major employment areas will provide a comparison across the United States, and possibly internationally. The results might show any geo-social differences that may exist. The expectation is that the differences in national cultures would result in significantly different benchmarks between various countries, such as in Europe and the Far East.

An interesting area of study (suggested by Maslow's hierarchy) would be to determine the potential differences in the way people rate managers in geo-social areas that are routinely concerned with basic survival needs, such as Palestine and Israel, as opposed to people in countries that everyday survival is not a major concern.

Performance Predictability

Another suggested area of research involves administering the MPS longitudinally within individual companies. In order to accomplish this type of research, the researcher will need access to proprietary information. A longitudinal study may relate the changes in scores to specific changes in outcomes, such as profitability, turnover, production, and sales.

Questionnaire Design

Additional examination of the descriptive analysis and other statistical results of the various groupings of the individual questions from the sample data from this study could result in improving the effectiveness of the sample-survey instrument itself, by reducing the number of statements or re-arranging the groupings. However, the broad number of questions makes it a better tool for trainers and consultants for discussion in groups.

Scoring Scale

Instead of a 0 to 10 ranking scale, surveys could use a seven or eight point Likert scale. This change might eliminate any unknown bias in scoring, resulting from having only four positive rankings and seven negative rankings on the 0 to 10 scale used in this study. However, we have no indication that this scale results in any systematic bias.

Expectation Drivers

An intriguing research question remains unanswered in this study, "What are the fundamental drivers behind the specific employee expectations for all, or part, of the 160 statements within the eight survey/constructs?" For example, why is trust between workers and managers so important? Do all employees place the same emphasis on trusting their managers? Intuition tells us that individuals may differ greatly. However, future research may produce a cluster of drivers for each major area that appear more than any other motivating forces and may not be obvious on the surface. The results of this further qualitative research might assist management to understand more fully the challenges facing them in meeting their employees' expectations.

Conclusions and recommendations

Despite its recognized limitations, this study is worthwhile for several important reasons. Altmann (2000) discusses the benefits of surveying employees to discover their perceptions of the climate of the organization. He offers that companies that conduct climate surveys may gain one or more of the following outcomes; 1) better employee buy-in, 2) more positive work results, 3) improved communication, 4) comparisons to others in the industry, and 5) greater management involvement (Altmann, 2000). This study supports Altmann's ideas regarding the value of periodic employee surveys. Although not a performance evaluation in itself, the MPS includes numerous questions that can be used to illuminate certain non-numerical aspects of performance.

The results of this study reveal three specific aspects of the research:

1. The statements used in the self-assessment survey are valid indicators of the state of a company's managerial effectiveness and organizational behavior.

2. The sample results provide significant data for establishing an organizational behavior effectiveness grade which can be used by management to compare any given company's own individual overall self-assessment grade to a meaningful standard management effectiveness grade. We established such a grade for Southeast Louisiana. 3. The effectiveness scores given to specific statements, or group of statements, provide a company's top management with an insight as to the particular strengths and weaknesses of the company's overall management efforts.

For the business community, the value of this research will be seen when the sample-survey is used repeatedly within individual companies, and the resulting knowledge is used for improving the management of technology. In the United States, experience shows that managers do not routinely have to deal with their employees' survival needs. Under normal circumstances, hazardous working conditions and terrorists' attacks excluded, management does not usually have to worry about their workers' personal safety needs other than for normal on-the-job safety concerns. However, management needs to be vitally concerned about their employees' social and self-esteem needs on a daily basis.

The data gathered from the MPS tells a rather unique story and paints a vivid picture of the organization. It describes the managerial and social atmosphere in detail. The positive or negative feelings associated with an employee's personal organizational stories prompt the score he or she awards to each statement. If a company gets a low score in a certain area like trust and loyalty, there is a reason; and that low score generally indicates that there is an important story that needs to be examined. After conducting the MPS within a specific organization, management can re-visit those stories in an effort to find out why the employees feel about the company the way they do.

  Table VI, were well below what has now been established as the Mean and Standard Deviation benchmarks for Southeast Louisiana.

Table III :

 III Deviation from a normal distribution for the eight survey/constructs

	Construct	Construct Construct	Cronbach alpha Asymptotic sigma, 2 tailed for one-Cronbach's alpha
		1. Morale and Motivation 1. Morale and Motivation 0.9254 0.9482 sample K-S test assuming a normal
		2. Training and Evaluati on distribution 0.9444 2. Training and Evaluation 0.9073
	3. Human Relations 4. Communication 1. Morale and Motivation 3. Human Relations	0.546	0.9498 0.9719 0.9238
	5. Resources 2. Training and Evaluation 4. Communication	0.228	0.9687 0.9444
	3. Human Relations	6. Problem Solving 5. Resources	0.052	0.9627 0.9350
	4. Communication	7. Managerial Stress 6. Problem Solving	0.085	0.9432 0.9437
	5. Resources	8. Equality and Diversity 7. Managerial Stress	0.036	0.9642 0.9177
	6. Problem Solving	Total score: Total score:	0.148	0.9730 0.9363
	7. Managerial Stress	Table I: Internal reliability for the 2002 survey 0.126
	8. Equality and Diversity	0.207
	Total score:		0.211

Table IV :

 IV Internal Reliability for the 1995-1997 survey


Christophe Dutang
email: dutang@math.unistra.fr

Existence theorems for generalized Nash equilibrium problems: an analysis of assumptions

Keywords: Noncooperative games, Existence theorem, Nash equilibrium. MSC2000: 91A10

The generalized Nash equilibrium, where the feasible sets of the players depend on other players' action, becomes increasingly popular among academics and practitionners. In this paper, we provide a thorough study of theorems guaranteeing existence of generalized Nash equilibria and analyze the assumptions on practical parametric feasible sets.

Introduction

In noncooperative game theory, solution concepts had been searched for years at the beginning of the XX th century, cf. [START_REF] Neumann | Theory of Games and Economic Behavior[END_REF]. Thankfully, the Nobel prize laureate, John F. Nash, proposed a unified solution concept for noncooperative games, latter called Nash equilibrium, with [START_REF] Nash | Equilibrium points in N-person games[END_REF][START_REF] Nash | Non-cooperative games[END_REF]. Despite some critiscism, this solution concept is widely used among academics to model noncooperative behavior. Classical applications of Nash equilibrium include computer science, telecommunication, energy markets, and many others, see [START_REF] Facchinei | Generalized Nash equilibrium problems[END_REF] for a recent survey. In this note, we focus on noncooperative games with infinite action space and one-period horizon. Let be N the number of players. The strategy set of player i is denoted by X i ⊂ R ni and the payoff function by θ i : X → R (to be maximized), where X = X 1 × • • • × X N . Player i's (pure) strategy is denoted by x i ∈ X i while x -i ∈ X -i denotes the other players' action, i.e. x -i = (x 1 , . . . , x i-1 , x i+1 , . . . , x N ) and

X -i = X 1 × • • • × X i-1 × X i+1 × • • • × X N .
A game is thus described by (N, X i , θ i (.)). Definition 1.1. A Nash equilibrium is a strategy point x ∈ X such that no player has an incentive to deviate, i.e. for all i ∈ {1, . . . , N },

∀x i ∈ X i , θ i (x i , x -i ) ≤ θ i (x i , x -i ). (1)
Originally, Nash introduced the equilibrium concept to finite games in [START_REF] Nash | Equilibrium points in N-person games[END_REF][START_REF] Nash | Non-cooperative games[END_REF], i.e. X i is a finite set. Therefore, he used the mixed strategy concept (i.e. a probability distribution over the pure strategies) and proved the existence of such an equilibrium in that context. We report here the existence theorem of [START_REF] Nikaido | Note on non-cooperative convex games[END_REF] for infinite games.

Theorem 1.2 (Nash). Let N agents be characterized by an action space X i and an objective function θ i . If ∀i ∈ {1, . . . , N }, X i is nonempty, convex and compact; θ i : X → R is continuous with X = X 1 × • • • × X N and ∀x -i ∈ X -i , x i → θ i (x i , x -i ) is concave on X i , then there exists a Nash equilibrium.

The concavity assumption of the objective function θ i with respect to x i is sometimes called playerconcavity. When dealing with cost functions rather payoff functions, the concavity assumption has to be replaced by a convexity assumption. Most existence theorems of Nash equilibrium rely on a fixed-point argument, and this from the very beginning. Indeed when Nash introduced his equilibrium concept, a fixed-point theorem is used: the Kakutani theorem in [START_REF] Nash | Equilibrium points in N-person games[END_REF] and the Brouwer theorem in [START_REF] Nash | Non-cooperative games[END_REF].

Since the introduction of games (N, X i , θ i (.)), many extensions have been proposed in the literature: discontinuous payoffs (e.g. [START_REF] Dasgupta | The existence of equilibrium in discontinuous economic games, i:theory[END_REF]), non concave payoffs (e.g. [START_REF] Baye | Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs[END_REF]), topological action spaces (e.g. [START_REF] Lu | On the existence of pure-strategy Nash equilibrium[END_REF][START_REF] Park | A variant of the Nash equilibrium theorem in generalized convex spaces[END_REF]), constrained strategy sets (e.g. [START_REF] Debreu | A social equilibrium existence theorem[END_REF][START_REF] Rosen | Existence and uniqueness of equilibrium points for concave N-person games[END_REF]). In the following, we consider the latter extension dealing with games where each player has a range of actions which depends on the actions of other players. This new extension leads to the so-called generalized Nash equilibrium. Let 2 Xi be the family of subsets of X i . Let C i : X -i → 2 Xi be constraint correspondence of Player i, i.e. a function mapping a point in X -i to a subset of X i . Thus, C i (x -i ) defines the ith player action space given other players' action x -i . Typically, the constraint correspondence C i is defined by a parametrized action space as

C i (x -i ) = {x i ∈ X i , g i (x i , x -i ) ≥ 0}
, where g i : X → R mi is a constraint function. When g i does not depend on x -i , we get back to standard game. A generalized game is described by (N, X i , C i (.), θ i (.)) and is also called an abstract economy in reference to Debreu's economic work [START_REF] Arrow | Existence of an equilibrium for a competitive economy[END_REF][START_REF] Debreu | A social equilibrium existence theorem[END_REF]. Definition 1.3. The generalized Nash equilibrium for a generalized game (N, X i , C i , θ i ) is defined as a point x solving for all i ∈ {1, . . . , N },

x i ∈ arg max xi∈Ci(x -i ) θ i (x i , x -i ). (2)
In the present paper, we provide a self-contained survey of existence theorems for generalized Nash equilibrium. We also emphasize the use of fixed-point theorems in the proof of such theorems. A second purpose of this paper is to analyze the assumptions of those theorems on practical applications, and in particular the assumption on the constraint correspondence. Now, we set the outline of this paper. Section 2 gives the minimum required mathematical tools to study generalized Nash equilibria. Then, Section 3 presents the most recent existence theorems. Finally, Section 4 focuses on the analysis of assumptions when dealing with parametrized constrained sets.

Preliminaries

The mathematical tools needed are briefly summarized in this section, so that this paper is self-contained. For further details, we refer readers to the following books [START_REF] Aubin | Optima and Equilibria: An Introduction to Nonlinear Analysis[END_REF][START_REF] Border | Fixed Point Theorems with Applications to Economics and Game Theory[END_REF][START_REF] Ewald | Games, fixed points and mathematical economics[END_REF][START_REF] Ichiishi | Game Theory for Economic Analysis[END_REF][START_REF] Ok | Real Analysis with Economic Applications[END_REF]. In the following, X and Y are two metric spaces.

Quasiconcavity

Refinements of concavity are first presented with characterizations based directly on the function f . Special characterizations when f is continuously differentiable or twice continuously differentiable exists but are omitted here, see [START_REF] Diewert | Nine kinds of quasiconcavity and concavity[END_REF] for a comprehensive study.

Definition 2.1. A function f : X → Y is concave (resp. convex) iff ∀x, y ∈ X, ∀λ ∈ [0, 1], f (λx + (1 -λ)y) ≥ (resp. ≤) λf (x) + (1 -λ)f (y). ( 3
)
Strict convexity/concavity is obtained when Inequality (3) is strict.

Concavity of f can be also be defined in terms of the graph of f . Let hyp(f ), epi(f ) be the hypograph and the epigraph of f , defined as hyp(f ) = {(x, y), y ≥ f (x)} and epi(f ) = {(x, y), y ≤ f (x)}. The concavity (resp. convexity) of a function f is equivalent to the convexity of hyp(f ) (resp. epi(f )). So, it is immediate that hyp(min(f 1 , f 2 )) = hyp(f 1 ) ∩ hyp(f 2 ): an intersection of two convex sets (resp. epi(max(f 1 , f 2 )) = epi(f 1 ) ∩ epi(f 2 )). The quasiconcavity is now introduced by relaxing Inequality (3).

Definition 2.2. A function

f : X → Y is quasiconcave (resp. quasiconvex) iff ∀x, y ∈ X, ∀λ ∈]0, 1[, f (λx + (1 -λ)y) ≥ min(f (x), f (y)), resp. f (λx + (1 -λ)y) ≤ max(f (x), f (y)). ( 4
)
Again, strict quasiconvexity/concavity is obtained when Inequality (4) is strict.

A univariate quasiconvex (resp. quasiconcave) function is either monotone or unimodal. Obviously convexity implies quasiconvexity. To better catch the meaning of quasi-concavity in contrast to concavity, we plot on Figure 1 examples of a concave function, a non-concave quasi-concave function and a non-quasiconcave function.

0 2 4 6 8 10
-30

-20 -10 -5 0 concave, quasi-concave f(x)=-(x-4)^2+3 x f(x) 0 2 4 6 8 10
0 1 2 3 4 5 6 not concave, quasi-concave g(x)=atan(x)*10^7/(x+10)^6 x g(x) 0 2 4 6 8
-200

-150

-100

-50 0 not concave, not quasi-concave h(x)=-(x-2)^2*(x-7)^2 x h(x)

Correspondences

As unveiled in the introduction, new tools are used to refine the action strategy set from a compact set in Equation ( 1) to a player-dependent constrained set in Equation ( 2). Thus, correspondences, also called multi-valued functions, point-to-set maps or set-valued mappings, are introduced.

Definition 2.3. A correspondence F : X → 2 Y is an application such that ∀x ∈ X, F (x) is a subset of Y . A correspondence is also denoted by F : X → P(Y ) or F : X ⇒ Y . Given F , the domain is dom(F ) = {x ∈ X, F (x) = ∅}, the range is rg(F ) = x F (x) and the graph is Gr(F ) = {(x, y) ∈ X × Y, y ∈ F (x)}. Example 2.4. Typical examples of correspondences are the inverse of a function f , x → f -1 (x) (since f -1 (x) might be empty, a singleton and a set); a constraint set x → {x, f (x) ≤ c}; the generalized gradient x → ∂f (x) or F : x → [-|x|, |x|]. Definition 2.5. For a correspondence F : X → 2 Y , the image of a subset B by F is defined as F -1 (B) = {x ∈ X, F (x) = B}. The exterior image (also called upper inverse) is F + (B) = {x ∈ X, F (x) ⊂ B} whereas the interior image (also called lower inverse) is F -(B) = {x ∈ X, F (x) ∩ B = ∅}.
A type of continuity for set-valued mappings has been introduced by Bouligand and Kuratowski in 1932: the lower and the upper semicontinuity (abbreviated l.s.c. and u.s.c.). In the literature, there are two concurrent definitions: the semicontinuity in the sense of Berge (e.g. [6, page 109]) and the semicontinuity in the sense of Hausdorff (e.g. [3, page 38-39]). These two definitions depend on the property of the set F (x), yet, they are equivalent if F is compact-valued. In that case, the u.s.c./l.s.c. continuity can be defined on the exterior/interior images of F .

Definition 2.6. F : X → 2 Y is upper semicontinuous (u.s.c.) at x, iff ∀y ∈ Y , F + ({y}) is an open set in X. F : X → 2 Y is lower semicontinuous (l.s.c.) at x, iff ∀y ∈ Y , F -({y}) is an open set in X.
When studying parametrized constrained sets, it is generally more convenient to work with characterizations by sequences. Therefore, the equivalent definition of the semicontinuity in terms of sequences are now given, see e.g. [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF].

Definition 2.7. F : X → 2 Y is upper semicontinuous (u.s.c.) at x, iff for all sequence (x n ) n ∈ X, x n -→ n→+∞ x, ∀y n ∈ T (x n ) and ∀y ∈ Y , y n -→ n→+∞ y ⇒ y ∈ T (x). Definition 2.8. F : X → 2 Y is lower semicontinuous (l.s.c.) at x, iff for all sequence (x n ) n ∈ X, x n -→ n→+∞ x, ∀y ∈ T (x), there exists a sequence (y k ) k ∈ Y , such that y k -→ n→+∞ y and ∀k ∈ N, y k ∈ T (x k ). Example 2.9. Let F defined by F (x) = [-1, 1] if x = 0 or {0} otherwise. F is l.s.c. in 0 but not u.s.c. Let G defined by G(x) = {0} if x = 0 or [-1, 1] otherwise. G is u.s.c. in 0 but not l.s.c. Let H defined by H(x) = {0} if x = 0 or {-1, 1} otherwise. H is neither u.s.c. nor l.s.c. in 0.

Theorems for correspondences

Thirdly, the necessary fixed-point theorems (for correspondences) are given, namely the Kakutani theorem [START_REF] Kakutani | A generalization of Brouwer's fixed point theorem[END_REF] and the Begle theorem [START_REF] Begle | A fixed point theorem[END_REF]. Let us first recall by the Brouwer theorem that a continuous function f from a finite-dimensional ball into itself admits a fixed-point. The Kakutani theorem is a valuable extention of the Brouwer's theorem to correspondences and is reported from [START_REF] Aubin | Set-valued Analysis[END_REF]. The original theorem of [START_REF] Kakutani | A generalization of Brouwer's fixed point theorem[END_REF] does not have any ambiguity about upper semicontinuity, since the author works in a finite-dimensional space with compact-valued mapping.

Theorem 2.10 (Kakutani). Let K be a nonempty compact convex subset of a Banach space (e.g. R n ) and T : K → 2 K a correspondence. If T is u.s.c. such that ∀x ∈ K, T (x) is nonempty, closed and convex, then T admits a fixed-point theorem.

The Begle theorem ( [START_REF] Begle | A fixed point theorem[END_REF]) is an extension of a fixed-point theorem for locally connected spaces by [START_REF] Eilenberg | Fixed point theorems for multi-valued transformations[END_REF][START_REF] Lefschetz | On the fixed point formula[END_REF], which in turn extends the Brouwer fixed-point theorem. To introduce this theorem, contractible polyhedrons have first to be defined: contractibility is in a sense related to convexity, see e.g. [START_REF] Kuhn | Contractibility and convexity[END_REF]. Definition 2.11. A geometric polyhedron is a finite union of convex hulls of finite-point sets.

Definition 2.12. A polyhedron is a subset S of R n homeomorphic to a geometric polyhedron P , i.e. there exists a bijective function between S and P . Definition 2.13. Contractible sets are nonempty sets deformable into a point by a continuous function (homotopy).

Example 2.14. Any star domain of Euclidean spaces is contractible whereas a finite-dimensional sphere is not. Any convex set of Euclidean spaces is contractible.

The Begle theorem reported here is the version from [START_REF] Debreu | A social equilibrium existence theorem[END_REF], originally contractible sets are replaced by absolute retracts in [START_REF] Begle | A fixed point theorem[END_REF].

Theorem 2.15 (Begle). Let Z be a contractible polyhedron and φ : Z → 2 Z be upper semicontinuous. If ∀z ∈ Z, φ(z) is contractible, then φ admits a fixed point.

Finally, a last theorem needed is the Berge's maximum theorem, see e.g. [29, page 229] and [7, page 64].

Theorem 2.16 (Berge's maximum theorem). Let X, Y be two metric spaces, f : X × Y → R be an objection function and F : X → 2 Y a constraint correspondence. Assume that f is continuous, F is both l.s.c. and u.s.c.; and F is nonempty and compact valued. Then we have

(i) φ : x → max y∈F (x) f (x, y) is a continuous function from X in R. (ii) Φ : x → arg max y∈F (x) f (x, y) is u.s.c. correspondence from X in 2 Y and compact-valued.
As shown in the proof of Theorem 3.1, if in addition f is quasiconcave in y, then Φ is convex valued. Note that Φ(x) is sometimes written {y ∈ F (x), f (x, y) = φ(x)}. The sequel demonstrates that the maximum theorem and the two fixed-point theorems 2.10 and 2.15 are the base recipes for showing the existence of a generalized Nash equilibrium.

State-of-the-art existence theorems

Showing the existence of a generalized Nash equilibrium can be tackled in two different ways: either a direct approach based on fixed-point theorems or a reformulation based on quasi-variational inequalities. Proofs are given to emphasize how the maximum theorem is the link between optimization subproblem (2) and fixed-point theorems.

The direct approach

Firstly, we investigate the direct approach. Theorem 3.1 was established by [START_REF] Arrow | Existence of an equilibrium for a competitive economy[END_REF] in the context of abstract economy, so a simplified version by [START_REF] Ichiishi | Game Theory for Economic Analysis[END_REF] is reported below. Some equivalent reformulations of Theorem 3.1 using a preference correspondence rather than a payoff function are also available in the following books: Theorem 19.8 in [START_REF] Border | Fixed Point Theorems with Applications to Economics and Game Theory[END_REF] and Theorem 3.7.1 in [START_REF] Ewald | Games, fixed points and mathematical economics[END_REF]. Note that [START_REF] Aubin | Optima and Equilibria: An Introduction to Nonlinear Analysis[END_REF] propose a different version, called the Arrow-Debreu-Nash theorem, where objective functions are player-concave rather than player-quasiconcave. Theorem 3.1. Let N players be characterized by an action space X i , a constraint correspondence C i and an objective function θ i : X → R. Assume for all players, we have (i) X i is nonempty, convex and compact subset of a Euclidean space, (ii) C i is both u.s.c. and l.s.c. in

X -i , (iii) ∀x -i ∈ X -i , C i (x -i ) is nonempty, closed, convex, (iv) θ i is continuous on the graph Gr(C i ) 1 , (v) ∀x ∈ X, x i → θ i (x i , x -i ) is quasiconcave on C i (x -i ),
Then there exists a generalized Nash equilibrium.

Proof. Since θ i is continuous, C i is both l.s.c. and u.s.c.; and C i is nonempty and compact valued, the maximum theorem implies that the best response correspondence defined as

x -i Pi -→ arg max xi∈Ci(x-i) θ i (x i , x -i )
is u.s.c. and compact valued. Furthermore, as θ i is player quasiconcave, P i is convex valued. Let z i , y i ∈ P i (x -i ). By definition of maximal points, ∀x i ∈ C i (x -i ), we have θ i (y i , x -i ) ≥ θ i (x i , x -i ) and θ i (z i , x -i ) ≥ θ i (x i , x -i ). Let λ ∈]0, 1[. By the quasiconcaveness assumption, we get

θ i (λy i + (1 -λ)z i , x -i ) ≥ min (θ i (y i , x -i ), θ i (z i , x -i )) ≥ θ i (x i , x -i ).
Hence, λy i + (1 -λ)z i ∈ P i (x -i ), i.e. P i (x -i ) is a convext set. Furthermore, P i is also nonempty valued since C i (x -i ) is nonempty. Now, consider the Cartesian product of P i (x -i ) to define Φ as

Φ : X → 2 X1 × • • • × 2 X N x → P 1 (x -1 ) × • • • × P N (x -N )
where X is a subset of R n with n = i n i . This multiplayer best response is nonempty, convex and compact valued. In our finite-dimensional setting and with a finite Cartesian product, the upper semicontinuity of each component P i implies the upper semicontinuity of Φ, see Prop 3.6 of [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF]. Finally, the Kakutani theorem gives the existence result.

The Debreu theorem ( [START_REF] Debreu | A social equilibrium existence theorem[END_REF]) based on contractible sets is now given. Originally, the upper-semicontinuity is replaced by the closedness of the graph Gr(C i ), but this is equivalent since contractible sets are closed and compact sets. Theorem 3.2. Let N agents be characterized by an action space X i and X = X 1 × • • • × X N . Let a payoff function θ i : X → R and a restricted action space C i (x -i ) given other player actions x -i . Each agent i maximizes its payoff on C i (x -i ). If for all agents, we have

(i) X i is a contractible polyhedron, (ii) C i : X -i → 2 Xi is u.s.c., (iii) θ i is continuous from Gr(C i ) to R, (iv) φ i : x -i → max xi∈Ci(x-i) θ i (x i , x -i ) is continuous, (v) ∀x -i ∈ X -i , the best response set M x-i = {x i ∈ X i (x -i ), θ i (x i , x -i ) = φ i (x -i )} is contractible,
Then there exists a generalized Nash equilibrium.

Proof. Let G i = Gr(C i ). Again, we work on the best response set, which is defined as

M i = {(x i , x -i ) ∈ X i × X -i , x i ∈ M x-i } = {(x i , x -i ) ∈ G i , θ i (x i , x -i ) = φ i (x -i )}.
This set is closed since the functions φ i and θ i are continuous and c i is u.s.c.. Let Φ be the correspondence defined as Φ(x) = M x-1 × • • • × M x -N . Using the Cartesian product, the graph of Φ is given by

Gr(Φ) = {(x, y) ∈ X × X, y ∈ Φ(x)} = N i=1 {(x, y) ∈ X × X, (y i , x -i ) ∈ M i },
a finite intersection of closed sets. As Gr(Φ) is closed, Φ is u.s.c.. Moreover for all x, Φ(x) is contractible as a finite Cartesian product of contractible sets. Applying the Begle fixed-point theorem completes the proof.

The QVI reformulation

The generalized Nash equilibrium problem (2) can be reformulated in the quasi-variational inequality (QVI) framework. Variational inequality and QVI are first described and then the existence theorem is given. Definition 3.3 (Variational Inequality). Given a function F : R n → R n and a set K ⊂ R n a Variational Inequality, denoted by V I(K, F (.)), is to find a vector x such that x ∈ K and ∀y ∈ K, (y -x ) T F (x ) ≥ 0. Definition 3.4 (Quasi-Variational Inequality). Given a function F : R n → R n and a correspondence K : R n → 2 R n , a Quasi-Variational Inequality, denoted by QV I(K(.), F (.)), is to find a vector x such that x ∈ K(x ) and ∀y ∈ K(x ), (y -x

) T F (x ) ≥ 0.
For examples of applications of variational inequality problems and links with optimization, see e.g. [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementary Problems[END_REF][START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementary Problems[END_REF]. Definition 1.3 can be reformulated as the Quasi-Variational Inequality problem QV I(C(.), Θ(.)) with

C(x) = C 1 (x -1 ) × • • • × C N (x -N ) and Θ(x) =    ∇ x1 θ 1 (x)
. . .

∇ x N θ N (x)    , (5)
see e.g. [START_REF] Harker | Generalized Nash games and quasi-variational inequalities[END_REF] and [START_REF] Facchinei | Generalized Nash equilibrium problems[END_REF] for a proof. Note that this reformulation assumes the differentiability of objective function θ i . An existence theorem based on the QVI approach developped in [START_REF] Fukushima | Quasi-variational inequalities, generalized Nash equilibria, and multileader-follower games[END_REF] is now given.

Theorem 3.5. Let N players be characterized by an action space X i , a constraint correspondence C i and an objective function θ i : X → R. Assume for all players, θ i is continuously differentiable on the graph Gr(C i ).

Let C(x) = C 1 (x -1 ) × • • • × C N (x -N ). Assume there exists a compact convex subset T ⊂ R n , (i) ∀x ∈ T , C(x) is nonempty, closed, convex subset of T , (ii) C is both u.s.c. and l.s.c. in T ,
Then there exists a generalized Nash equilibrium.

Proof. Using Θ as given in Equation ( 5), the correspondence F : T → 2 T is defined as

F (x) = arg max z∈C(x) -(z -x) T Θ(x).
Let x be a fixed-point of F . We have

x ∈ F (x ) ⇔ x ∈ arg max z∈C(x ) -(z -x ) T Θ(x ) ⇔ ∀z ∈ C(x ), -(z -x ) T Θ(x ) ≤ -(x -x ) T Θ(x ) = 0 ⇔ ∀z ∈ C(x ), (z -x ) T Θ(x ) ≥ 0.
Thus, the QVI reformulation of (2) turns out to be a fixed-point problem. By assumption, C is nonempty, compact valued and both u.s.c. and l.s.c.. The function (z, y) → -(z -y) T Θ(y) is continuous since θ i is continuously differentiable. Therefore by the maximum theorem, the correspondence F is u.s.c. and compact valued. Furthermore, the function z → -(z -x) T Θ(x) is a linear function (hence convex). So, F is also convex-valued (by the maximum theorem), hence F (x) is a contractible set for all x ∈ T . Applying the Begle fixed-point theorem completes the proof.

A significative part of games are such that player strategies are required to satisfy a common coupling constraint (such games are called jointly convex games) see [START_REF] Facchinei | Generalized Nash equilibrium problems[END_REF] and the references therein. In jointly convex games, the contraint correspondence simplifies to

C i : X -i → 2 Xi x -i → {x i ∈ X i , (x i , x -i ) ∈ K}, (6)
where

K ⊂ X 1 × • • • × X N is a nonempty convex set.
We are now interested in points solving the (classical) variational inequality problem V I(K, Θ(.)) with K given in Equation ( 6) and Θ given in Equation ( 5). As expected, not all solutions of the generalized Nash equilibrium (2) (i.e. solutions of QV I(C(.), Θ(.))) solves this variational inequality problem. Therefore, a special type of generalized Nash equilibrium has been introduced: a variational equilibrium also called a normalized equilibrium. A variational equilibrium has a special interpretation in terms of Lagrange multipliers of the corresponding KKT systems of the GNEP, see e.g. [START_REF] Facchinei | On generalized Nash games and variational inequalities[END_REF][START_REF] Fukushima | Quasi-variational inequalities, generalized Nash equilibria, and multileader-follower games[END_REF].

Definition 3.6 (Variational equilibrium). A strategy x is a variational equilibrium of a generalized game (N, X i , C i (.), θ i (.)) if x solves V I(K, Θ(.)) with K given in Equation ( 6) and Θ given in Equation ( 5).

Theorem 3.7. Let N players be characterized by an action space X i , a constraint correspondence C i and an objective function θ i : X → R. Assume for all players, θ i is continuously differentiable on the graph Gr(C i ) and there exists a nonempty convex compact set K ⊂ R n such that C i (x -i ) = {x i ∈ X i , (x i , x -i ) ∈ K}, then there exists a variational equilibrium.

Proof. For ease of notation, the subscript i is removed, x i and x -i are denoted by x and w, respectively. Let f be the function f (x, w) = min(g 1 (x, w), . . . , g m (x, w)). which is continuous by the continuity of g j . By the concavity with respect to x, f is also concave with respect to x. The upper level lev ≥0 f is the graph of C. By the continuity of f , the graph Gr(f ) is closed.

lev ≥0 f = {(x, w), f (x, w) ≥ 0} = {(x, w), ∀i = 1, . . . , m, g i (x, w) ≥ 0} = {(x, w), x ∈ C(w)} = Gr(C).
So C is u.s.c.. The level set of a convex function is also convex, so is lev ≥0 f (., w) with respect to x for all w. As f is continuous and g(x, w) > 0, there exists an open set O, such that ∀w ∈ O, f (x, w) > 0.

Since the upper level set lev ≥0 f (., w) for any w is convex and f is continuous, the interior intC(w) is nonempty. This guarantees the lower semicontinuity of C at w. Indeed, for all w ∈ O and all x ∈ intC( w), by the continuity of f and the assumption at (x, w), there exists a neighborhood W ⊂ O × intC( w) such that f is strictly positive on W , and also W ⊂ gph(C). As w → w, then certainly x belongs to the inner limit of C(w). This inner limit is a closed set, and so includes intC( w) ⊃ cl(intC( w)). Since C( w) is a closed convex set with noempty interior, cl(intC( w)) = C( w). Hence, the inner limit of C(w) contains C( w), i.e. C is l.s.c. at w by Theorem 5.9 of [START_REF] Rockafellar | Variational Analysis[END_REF].

The previous property is also given in Theorem 12 of [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF] and proved using the sequence characterization of semicontinuity. Theorem 13 of [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF] is reported here as it gives weaker conditions for the correspondence to be lower semicontinuous than Theorem 12. Proposition 4.3. Let C i : X -i → 2 Xi be the feasible set mapping as defined above. Let C i be the correspondence

C i (x -i ) = {x i ∈ X i , g i (x i , x -i ) > 0}. If each component g ij is lower semicontinuous (i.e. closedness of the epigraph) on x-i × C i (x -i ) and C i (x -i ) ⊂ cl( C i (x -i )), then C i is lower semicontinuous at x-i .
Proof. For ease of notation, the subscript i is removed, x i and x -i are denoted by x and w, respectively. If C( w) = ∅, then by assumption, C( w) = ∅ and the conclusion is trivial. Otherwise, when C( w) = ∅, we choose x ∈ C( w) and w n → w. Since C( w) ⊂ cl( C( w)), there exists a sequence (x m ) m of elements in C( w) such that x m → x. Construct the sequence n m such that n 0 = 0 and n m = max(n m-1 + 1, arg min k (∀l ≥ k, g(w l , x m ) > 0)). The sequence is well defined by the lower semicontinuity of g. Furthermore, the sequence (x nm ) m≥0 is such that x nm ∈ C(w nm ) with x nm → x, w nm → w and x ∈ C( w), i.e. C is l.s.c. at w. Continuous selections introduced by [START_REF] Michael | Continuous selections. 1[END_REF] can be used to further relaxed assumptions on g i . Originally, [START_REF] Michael | Continuous selections. 1[END_REF] works with topological spaces and uses the Berge's semicontinuity. Their Proposition 2.3 is reported below.

Proposition 4.4. If φ : X → 2 Y is l.s.c. and φ : X → 2 Y such that for every x ∈ X, cl(φ(x)) = cl(ψ(x)), then ψ is also l.s.c.. Property 4.4 has strong consequences on the lower semicontinuity of the correspondence C i and justifies the [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF]'s approach to use the correspondence C i rather than C i , since images have the same closure set. Therefore, the lower semicontinuity of each component g ij suffices to get the lower semicontinuity of C i . With the continuity of g ij , it is even more straightforward to see that for all x i ∈ C i (x -i ), there exists a sequence (x -i,n ) n and x i,n ∈ C i (x -i,n ) such that for all n ≥ n 0 , g i (x i,n , x -i,n ) > 0. Other types of conditions not based on strict inequalities are given in [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF] .

The nonemptyness, the closedness and the convexity

Finally, we turn our attentions to other assumptions. Theorems 3.1, 3.2 and 3.5 also require C i to be nonempty, convex and closed valued. The convexity assumption on C i (x -i ) is satisfied when g i is quasiconcave with respect to x i . This is not immediate with the definition of quasi-concavity given in Section 2. But an equivalent definition for a function f to be quasiconcave is that all upper level sets U f (r) = {x ∈ X, f (x) ≥ r} are convex for all r, see [START_REF] Diewert | Nine kinds of quasiconcavity and concavity[END_REF]. Thus, if g i is quasiconcave, then U gi (0) is convex. The nonemptyness assumption is the most challenging assumption. Except to have a strict inequality condition and the continuity of g i 's, it is hard to find general conditions. Finally, the closedness assumption is satisfied when g i 's are continuous.

Figure 1 :

 1 Figure 1: Examples and counter-examples of quasi-concavity

There is no need for θ i to be continuous on the whole space X, since only feasible points (i.e. those in Gr(C i )) matters in Equation (2).

Proof. Same proof as Theorem 3.5 with C(x) replace by K which has the same properties.

Parametrized constrained sets

This final section aims to provide criteria to guarantee the assumptions of previous theorems, as well as, proofs for such criteria. For this purpose, a parametrized constraint set is considered

for x -i ∈ X -i , where

A central assumption of Theorems 3.1, 3.2 and 3.5 is to require C i to be both lower and upper semicontinuous. Yet, Theorem 3.2 requires C i to be u.s.c. and φ to be continuous, by Berge's maximum theorem, a sufficient condition is that C i is also l.s.c.. Other assumptions of these theorems are nonemptyness, convexity and closedness of C i (x -i ). Theorem 3.2 also requires X i and M i to be contractible: a sufficient condition is X i to be a convex and θ i to be player-quasiconcave.

The upper-semicontinuity

[31] devote a full chapter on set-valued analysis. Despite they formulate the outer and inner semicontinuity through superior limit of sets, they work with the Berge's semicontinuity (respectively the lower and upper semicontinuity). Their Theorem 5.7 of [START_REF] Rockafellar | Variational Analysis[END_REF] gives equivalent reformulations of upper semicontinuity using the graph properties, for which their Example 5.10 is a direct application. A small variant is given here by removing equality constraints. Proposition 4.1. Let C i : X -i → 2 Xi be the feasible set mapping defined in Equation [START_REF] Border | Fixed Point Theorems with Applications to Economics and Game Theory[END_REF]. Assume X i ⊂ R ni is closed and all components g ij 's are continuous on X i × X -i ⊂ R n , then the correspondence C i is u.s.c. on X -i .

Proof. For all j = {1, . . . , m}, by the continuity of the jth component g ij , the set of

A weaker assumption on the constraint function g i is given in Theorem 10 of [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF]. [START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF] only assumes that each component g ij is an upper semicontinuous function (i.e. the closedness of the hypograph g ij ).

The lower-semicontinuity

Generally, conditions on g i in order that the correspondence is l.s.c. are harder to find. Nevertheless, [START_REF] Rockafellar | Variational Analysis[END_REF][START_REF] Hogan | Point-to-set maps in mathematical programming[END_REF] provide conditions for it. An application of Theorem 5.9 of [START_REF] Rockafellar | Variational Analysis[END_REF] to the constraint correspondence in Equation ( 7) is presented. Proposition 4.2. Let C i : X -i → 2 Xi be the feasible set mapping defined in Equation [START_REF] Border | Fixed Point Theorems with Applications to Economics and Game Theory[END_REF]. Assume g ij 's are continuous and concave in x i for each x -i . If there exists x such that g i (x i , x-i ) > 0 for all i, then C i is l.s.c. at x-i and in some neighborhood of x-i (and also u.s.c.).