Introduzione alla teoria dei giochi. Riflessioni didattiche

In questo elaborato si vuole introdurre la teoria dei giochi da un punto di vista matematico attraverso la trattazione di alcuni argomenti di base: i giochi in forma estesa e a somma zero, l'equilibrio di Nash e i giochi cooperativi.

Three Essays in Spatial Economics

This work contributes to the field of spatial economics by embracing three distinct modelling approaches, belonging to different strands of the theoretical literature. In the first chapter I present a theoretical model in which the changes in urban system’s degree of functional specialisation are linked to (i) firms’ organisational choices and firms’ location decisions. The interplay between firms’ internal communication/managing costs (between headquarters and production plants) and the cost of communicating with distant business services providers leads the transition process from an “integrated” urban system where each city hosts every different functions to a “functionally specialised” urban system where each city is either a primary business center (hosting advanced business services providers, a secondary business center or a pure manufacturing city and all this city-types coexist in equilibrium.The second chapter investigates the impact of free trade on welfare in a two-country world modelled as an international Hotelling duopoly with quadratic transport costs and asymmetric countries, where a negative environmental externality is associated with the consumption of the good produced in the smaller country. Countries’ relative sizes as well as the intensity of negative environmental externality affect potential welfare gains of trade liberalisation. The third chapter focuses on the paradox, by which, contrary to theoretical predictions, empirical evidence shows that a decrease in international transport costs causes an increase in foreign direct investments (FDIs). Here we propose an explanation to this apparent puzzle by exploiting an approach which delivers a continuum of Bertrand- Nash equilibria ranging above marginal cost pricing. In our setting, two Bertrand firms, supplying a homogeneous good with a convex cost function, enter the market of a foreign country. We show that allowing for a softer price competition may indeed more than offset the standard effect generated by a decrease in trade costs, thereby restoring FDI incentives.

Tecniche regionali di stima della portata di progetto: Impatto della correlazione spaziale tra le serie di osservazioni

La stima degli indici idrometrici in bacini non strumentati rappresenta un problema che la ricerca internazionale ha affrontato attraverso il cosiddetto PUB (Predictions on Ungauged Basins – IAHS, 2002-2013). Attraverso l’analisi di un’area di studio che comprende 61 bacini del Sud-Est americano, si descrivono e applicano due tecniche di stima molto diverse fra loro: il metodo regressivo ai Minimi Quadrati Generalizzati (GLS) e il Topological kriging (TK). Il primo considera una serie di fattori geomorfoclimatici relativi ai bacini oggetto di studio, e ne estrae i pesi per un modello di regressione lineare dei quantili; il secondo è un metodo di tipo geostatistico che considera il quantile come una variabile regionalizzata su supporto areale (l’area del bacino), tenendo conto della dislocazione geografica e l’eventuale struttura annidata dei bacini d’interesse. L’applicazione di questi due metodi ha riguardato una serie di quantili empirici associati ai tempi di ritorno di 10, 50, 100 e 500 anni, con lo scopo di valutare le prestazioni di un loro possibile accoppiamento, attraverso l’interpolazione via TK dei residui GLS in cross-validazione jack-knife e con differenti vicinaggi. La procedura risulta essere performante, con un indice di efficienza di Nash-Sutcliffe pari a 0,9 per tempi di ritorno bassi ma stazionario su 0,8 per gli altri valori, con un trend peggiorativo all’aumentare di TR e prestazioni pressoché invariate al variare del vicinaggio. L’applicazione ha mostrato che i risultati possono migliorare le prestazioni del metodo GLS ed essere paragonabili ai risultati del TK puro, confermando l’affidabilità del metodo geostatistico ad applicazioni idrologiche.

Teorema di Brouwer ed applicazioni

Lo scopo di questa tesi è analizzare il teorema del punto fisso di Brouwer, e lo faremo da più punti di vista, generalizzandolo e dando una piccola illustrazione di una sua possibile applicazione nella teoria dei giochi. Il teorema del punto fisso è uno dei teoremi prìncipi della topologia algebrica. Nella versione classica esso afferma che qualsiasi funzione continua che porta la palla unitaria di \R^{n} in se stessa possiede un punto fisso.

Mortality associated with discordant responses to antiretroviral therapy in resource-constrained settings

Objectives: We assessed mortality associated with immunologic and virologic patterns of response at 6 months of highly active antiretroviral therapy (HAART) in HIV-infected individuals from resource-limited countries in Africa and South America. Methods: Patients who initiated HAART between 1996 and 2007, aged 16 years or older, and had at least 1 measurement (HIV-1 RNA plasma viral load or CD4 cell count) at 6 months of therapy (3-9 month window) were included. Therapy response was categorized as complete, discordant (virologic only or immunologic only), and absent. Associations between 6-month response to therapy and all-cause mortality were assessed by Cox proportional hazards regression. Robust standard errors were calculated to account for intrasite correlation. Results: A total of 7160 patients, corresponding to 15,107 person-years, were analyzed. In multivariable analysis adjusted for age at HAART initiation, baseline clinical stage and CD4 cell count, year of HAART initiation, clinic, occurrence of an AIDS-defining condition within the first 6 months of treatment, and discordant and absent responses were associated with increased risk of death. Conclusions: Similar to reports from high-income countries, discordant immunologic and virologic responses were associated with intermediate risk of death compared with complete and no response in this large cohort of HIV-1 patients from resource-limited countries. Our results support a recommendation for wider availability of plasma viral load testing to monitor antiretroviral therapy in these settings.

Mortality after failure of antiretroviral therapy in sub-Saharan Africa

Objective  To assess the outcome of patients who experienced treatment failure with antiretrovirals in sub-Saharan Africa. Methods  Analysis of 11 antiretroviral therapy (ART) programmes in sub-Saharan Africa. World Health Organization (WHO) criteria were used to define treatment failure. All ART-naive patients aged ≥16 who started with a non-nucleoside reverse transcriptase inhibitor (NNRTI)-based regimen and had at least 6 months of follow-up were eligible. For each patient who switched to a second-line regimen, 10 matched patients who remained on a non-failing first-line regimen were selected. Time was measured from the time of switching, from the corresponding time in matched patients, or from the time of treatment failure in patients who remained on a failing regimen. Mortality was analysed using Kaplan–Meier curves and random-effects Cox models. Results  Of 16 591 adult patients starting ART, 382 patients (2.3%) switched to a second-line regimen. Another 323 patients (1.9%) did not switch despite developing immunological or virological failure. Cumulative mortality at 1 year was 4.2% (95% CI 2.2–7.8%) in patients who switched to a second-line regimen and 11.7% (7.3%–18.5%) in patients who remained on a failing first-line regimen, compared to 2.2% (1.6–3.0%) in patients on a non-failing first-line regimen (P < 0.0001). Differences in mortality were not explained by nadir CD4 cell count, age or differential loss to follow up. Conclusions  Many patients who meet criteria for treatment failure do not switch to a second-line regimen and die. There is an urgent need to clarify the reasons why in sub-Saharan Africa many patients remain on failing first-line ART.

Spatio-temporal credit assignment in population learning

Learning by reinforcement is important in shaping animal behavior, and in particular in behavioral decision making. Such decision making is likely to involve the integration of many synaptic events in space and time. However, using a single reinforcement signal to modulate synaptic plasticity, as suggested in classical reinforcement learning algorithms, a twofold problem arises. Different synapses will have contributed differently to the behavioral decision, and even for one and the same synapse, releases at different times may have had different effects. Here we present a plasticity rule which solves this spatio-temporal credit assignment problem in a population of spiking neurons. The learning rule is spike-time dependent and maximizes the expected reward by following its stochastic gradient. Synaptic plasticity is modulated not only by the reward, but also by a population feedback signal. While this additional signal solves the spatial component of the problem, the temporal one is solved by means of synaptic eligibility traces. In contrast to temporal difference (TD) based approaches to reinforcement learning, our rule is explicit with regard to the assumed biophysical mechanisms. Neurotransmitter concentrations determine plasticity and learning occurs fully online. Further, it works even if the task to be learned is non-Markovian, i.e. when reinforcement is not determined by the current state of the system but may also depend on past events. The performance of the model is assessed by studying three non-Markovian tasks. In the first task, the reward is delayed beyond the last action with non-related stimuli and actions appearing in between. The second task involves an action sequence which is itself extended in time and reward is only delivered at the last action, as it is the case in any type of board-game. The third task is the inspection game that has been studied in neuroeconomics, where an inspector tries to prevent a worker from shirking. Applying our algorithm to this game yields a learning behavior which is consistent with behavioral data from humans and monkeys, revealing themselves properties of a mixed Nash equilibrium. The examples show that our neuronal implementation of reward based learning copes with delayed and stochastic reward delivery, and also with the learning of mixed strategies in two-opponent games.

Spatio-temporal credit assignment in population learning

Learning by reinforcement is important in shaping animal behavior. But behavioral decision making is likely to involve the integration of many synaptic events in space and time. So in using a single reinforcement signal to modulate synaptic plasticity a twofold problem arises. Different synapses will have contributed differently to the behavioral decision and, even for one and the same synapse, releases at different times may have had different effects. Here we present a plasticity rule which solves this spatio-temporal credit assignment problem in a population of spiking neurons. The learning rule is spike time dependent and maximizes the expected reward by following its stochastic gradient. Synaptic plasticity is modulated not only by the reward but by a population feedback signal as well. While this additional signal solves the spatial component of the problem, the temporal one is solved by means of synaptic eligibility traces. In contrast to temporal difference based approaches to reinforcement learning, our rule is explicit with regard to the assumed biophysical mechanisms. Neurotransmitter concentrations determine plasticity and learning occurs fully online. Further, it works even if the task to be learned is non-Markovian, i.e. when reinforcement is not determined by the current state of the system but may also depend on past events. The performance of the model is assessed by studying three non-Markovian tasks. In the first task the reward is delayed beyond the last action with non-related stimuli and actions appearing in between. The second one involves an action sequence which is itself extended in time and reward is only delivered at the last action, as is the case in any type of board-game. The third is the inspection game that has been studied in neuroeconomics. It only has a mixed Nash equilibrium and exemplifies that the model also copes with stochastic reward delivery and the learning of mixed strategies.

Correcting mortality for loss to follow-up: a nomogram applied to antiretroviral treatment programmes in sub-Saharan Africa

Background The World Health Organization estimates that in sub-Saharan Africa about 4 million HIV-infected patients had started antiretroviral therapy (ART) by the end of 2008. Loss of patients to follow-up and care is an important problem for treatment programmes in this region. As mortality is high in these patients compared to patients remaining in care, ART programmes with high rates of loss to follow-up may substantially underestimate mortality of all patients starting ART. Methods and Findings We developed a nomogram to correct mortality estimates for loss to follow-up, based on the fact that mortality of all patients starting ART in a treatment programme is a weighted average of mortality among patients lost to follow-up and patients remaining in care. The nomogram gives a correction factor based on the percentage of patients lost to follow-up at a given point in time, and the estimated ratio of mortality between patients lost and not lost to follow-up. The mortality observed among patients retained in care is then multiplied by the correction factor to obtain an estimate of programme-level mortality that takes all deaths into account. A web calculator directly calculates the corrected, programme-level mortality with 95% confidence intervals (CIs). We applied the method to 11 ART programmes in sub-Saharan Africa. Patients retained in care had a mortality at 1 year of 1.4% to 12.0%; loss to follow-up ranged from 2.8% to 28.7%; and the correction factor from 1.2 to 8.0. The absolute difference between uncorrected and corrected mortality at 1 year ranged from 1.6% to 9.8%, and was above 5% in four programmes. The largest difference in mortality was in a programme with 28.7% of patients lost to follow-up at 1 year. Conclusions The amount of bias in mortality estimates can be large in ART programmes with substantial loss to follow-up. Programmes should routinely report mortality among patients retained in care and the proportion of patients lost. A simple nomogram can then be used to estimate mortality among all patients who started ART, for a range of plausible mortality rates among patients lost to follow-up.

Genetic determinants of alcoholic liver disease

Alcoholic liver disease (ALD) accounts for the majority of chronic liver disease in Western countries. The spectrum of ALD includes steatosis with or without fibrosis in virtually all individuals with an alcohol consumption of >80 g/day, alcoholic steatohepatitis of variable severity in 10-35% and liver cirrhosis in approximately 15% of patients. Once cirrhosis is established, there is an annual risk for hepatocellular carcinoma of 1-2%. Environmental factors such as drinking patterns, coexisting liver disease, obesity, diet composition and comedication may modify the natural course of ALD. Twin studies have revealed a substantial contribution of genetic factors to the evolution of ALD, as demonstrated by a threefold higher disease concordance between monozygotic twins and dizygotic twins. With genotyping becoming widely available, a large number of genetic case-control studies evaluating candidate gene variants coding for proteins involved in the degradation of alcohol, mediating antioxidant defence, the evolution and counteraction of necroinflammation and formation and degradation of extracellular matrix have been published with largely unconfirmed, impeached or even disproved associations. Recently, whole genome analyses of large numbers of genetic variants in several chronic liver diseases including gallstone disease, primary sclerosing cholangitis and non-alcoholic fatty liver disease (NAFLD) have identified novel yet unconsidered candidate genes. Regarding the latter, a sequence variation within the gene coding for patatin-like phospholipase encoding 3 (PNPLA3, rs738409) was found to modulate steatosis, necroinflammation and fibrosis in NAFLD. Subsequently, the same variant was repeatedly confirmed as the first robust genetic risk factor for progressive ALD.

Compensation, Incentives and Risk: Economic Theory and Russian Practise

Mr. Pechersky set out to examine a specific feature of the employer-employee relationship in Russian business organisations. He wanted to study to what extent the so-called "moral hazard" is being solved (if it is being solved at all), whether there is a relationship between pay and performance, and whether there is a correlation between economic theory and Russian reality. Finally, he set out to construct a model of the Russian economy that better reflects the way it actually functions than do certain other well-known models (for example models of incentive compensation, the Shapiro-Stiglitz model etc.). His report was presented to the RSS in the form of a series of manuscripts in English and Russian, and on disc, with many tables and graphs. He begins by pointing out the different examples of randomness that exist in the relationship between employee and employer. Firstly, results are frequently affected by circumstances outside the employee's control that have nothing to do with how intelligently, honestly, and diligently the employee has worked. When rewards are based on results, uncontrollable randomness in the employee's output induces randomness in their incomes. A second source of randomness involves the outside events that are beyond the control of the employee that may affect his or her ability to perform as contracted. A third source of randomness arises when the performance itself (rather than the result) is measured, and the performance evaluation procedures include random or subjective elements. Mr. Pechersky's study shows that in Russia the third source of randomness plays an important role. Moreover, he points out that employer-employee relationships in Russia are sometimes opposite to those in the West. Drawing on game theory, he characterises the Western system as follows. The two players are the principal and the agent, who are usually representative individuals. The principal hires an agent to perform a task, and the agent acquires an information advantage concerning his actions or the outside world at some point in the game, i.e. it is assumed that the employee is better informed. In Russia, on the other hand, incentive contracts are typically negotiated in situations in which the employer has the information advantage concerning outcome. Mr. Pechersky schematises it thus. Compensation (the wage) is W and consists of a base amount, plus a portion that varies with the outcome, x. So W = a + bx, where b is used to measure the intensity of the incentives provided to the employee. This means that one contract will be said to provide stronger incentives than another if it specifies a higher value for b. This is the incentive contract as it operates in the West. The key feature distinguishing the Russian example is that x is observed by the employer but is not observed by the employee. So the employer promises to pay in accordance with an incentive scheme, but since the outcome is not observable by the employee the contract cannot be enforced, and the question arises: is there any incentive for the employer to fulfil his or her promises? Mr. Pechersky considers two simple models of employer-employee relationships displaying the above type of information symmetry. In a static framework the obtained result is somewhat surprising: at the Nash equilibrium the employer pays nothing, even though his objective function contains a quadratic term reflecting negative consequences for the employer if the actual level of compensation deviates from the expectations of the employee. This can lead, for example, to labour turnover, or the expenses resulting from a bad reputation. In a dynamic framework, the conclusion can be formulated as follows: the higher the discount factor, the higher the incentive for the employer to be honest in his/her relationships with the employee. If the discount factor is taken to be a parameter reflecting the degree of (un)certainty (the higher the degree of uncertainty is, the lower is the discount factor), we can conclude that the answer to the formulated question depends on the stability of the political, social and economic situation in a country. Mr. Pechersky believes that the strength of a market system with private property lies not just in its providing the information needed to compute an efficient allocation of resources in an efficient manner. At least equally important is the manner in which it accepts individually self-interested behaviour, but then channels this behaviour in desired directions. People do not have to be cajoled, artificially induced, or forced to do their parts in a well-functioning market system. Instead, they are simply left to pursue their own objectives as they see fit. Under the right circumstances, people are led by Adam Smith's "invisible hand" of impersonal market forces to take the actions needed to achieve an efficient, co-ordinated pattern of choices. The problem is that, as Mr. Pechersky sees it, there is no reason to believe that the circumstances in Russia are right, and the invisible hand is doing its work properly. Political instability, social tension and other circumstances prevent it from doing so. Mr. Pechersky believes that the discount factor plays a crucial role in employer-employee relationships. Such relationships can be considered satisfactory from a normative point of view, only in those cases where the discount factor is sufficiently large. Unfortunately, in modern Russia the evidence points to the typical discount factor being relatively small. This fact can be explained as a manifestation of aversion to risk of economic agents. Mr. Pechersky hopes that when political stabilisation occurs, the discount factors of economic agents will increase, and the agent's behaviour will be explicable in terms of more traditional models.

Tuberculosis after initiation of antiretroviral therapy in low-income and high-income countries

We examined the incidence of and risk factors for tuberculosis during the first year of highly active antiretroviral therapy in low-income (4540 patients) and high-income (22,217 patients) countries. Although incidence was much higher in low-income countries, the reduction in the incidence of tuberculosis associated with highly active antiretroviral therapy was similar: the rate ratio for months 7-12 versus months 1-3 was 0.48 (95% confidence interval, 0.36-0.64) in low-income countries and 0.36 (95% confidence interval, 0.26-0.50) in high-income countries. A low CD4 cell count at the start of therapy was the most important risk factor in both settings.

Discordant responses to potent antiretroviral treatment in previously naive HIV-1-infected adults initiating treatment in resource-constrained countries: the antiretroviral therapy in low-income countries (ART-LINC) collaboration

OBJECTIVES: To assess the frequency of and risk factors for discordant responses at 6 months on highly active antiretroviral therapy (HAART) in previously treatment-naive HIV patients from resource-limited countries. METHODS: The Antiretroviral Therapy in Low-Income Countries Collaboration is a network of clinics providing care and treatment to HIV-infected patients in Africa, Latin America, and Asia. Patients who initiated therapy between 1996 and 2004, were aged 16 years or older, and had a baseline CD4 cell count were included in this analysis. Responses were defined based on plasma viral load (PVL) and CD4 cell count at 6 months as complete virologic and immunologic (VR(+)IR(+)), virologic only (VR(+)IR(-)), immunologic only (VR(-)IR(+)), and nonresponse (VR(-)IR(-)). Multinomial logistic regression was used to assess the association between therapy responses and clinical and demographic variables. RESULTS: Of the 3111 patients eligible for analysis, 1914 had available information at 6 months of therapy: 1074 (56.1%) were VR(+)IR(+), 364 (19.0%) were VR(+)IR(-), 283 (14.8%) were (VR(-)IR(+)), and 193 (10.1%) were VR(-)IR(-). OF THE 3111 patients eligible for analysis, 1914 had available information at 6 months of therapy: 1074 (56.1%) were VRIR, 364 (19.0%) were VRIR, 283 (14.8%) were (VRIR), and 193 (10.1%) were VRIR. Compared with complete responders, virologic-only responders were older, had a higher baseline CD4 cell count, had a lower baseline PVL, and were more likely to have received a nonstandard HAART regimen; immunologic-only responders were younger, had a lower baseline CD4 cell count, had a higher baseline PVL, and were more likely to have received a protease inhibitor-based regimen. CONCLUSIONS: The frequency of and risk factors for discordant responses were comparable to those observed in developed countries. Longer follow-up is needed to assess the long-term impact of discordant responses on mortality in these resource-limited settings.

Alcoholic liver disease in the elderly

Although per capita alcohol consumption, and thus the prevalence of alcoholic liver disease, decreases generally with age in Europe and in the United States, recently an increase in alcohol consumption has been reported in individuals over 65 years. Reasons explaining this observation may include an increase in life expectancy or a loss of life partners and, thus, loneliness and depression. Although ethanol metabolism and ethanol distribution change with age, and an elderly person's liver is more susceptible to the toxic effect of ethanol, the spectrum of alcoholic liver diseases and their symptoms and signs is similar to that seen in patients of all ages. However, prognosis of alcoholic liver disease in the elderly is poor. In addition, chronic alcohol consumption may enhance drug associated liver disease and may also act as a cofactor in other liver diseases, such as viral hepatitis and nonalcoholic fatty liver disease.